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Bc608 module2

2013-02-26T19:32:03Z

Bc608: Created page with " Normal 0 false false false EN-GB X-NONE X-NONE ==Inorganic Computational Lab: Ben Chappell (CID:00513494)== ===Introduction=== In this lab, ''ab initio'' (..."

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==Inorganic Computational Lab: Ben Chappell (CID:00513494)==

===Introduction===

In this lab, ''ab initio'' ("from scratch") quantum mechanical calculations are utilised in order to gain an insight into the structure and bonding of Inorganic complexes. DFT calculations with the B3LYP functional are used exclusively throughout this lab, due to their wide applicability and extensive use in wider research.<ref name="DFT_B3LYP_WHY_USED">K. Burke, J. Werschnik and E.K.U. Gross, ''Journal of Chemical Physics'' 2005, '''123''', 123-131 {{DOI|10.1063/1.1904586}}</ref> The basis sets for the calculations are discussed in each section, with an initial discussion of basis sets provided in the examination of BH3.

Gaussian 09W is used to perform calculations, whilst GaussView v5.0 is used to create, edit and display files used and generated by Gaussian.

===BH3===

====Geometry Optimisation====

In the calculation methods discussed in this lab, atomic orbitals are modelled by Slater Type Orbitals (STO), which show a dependance upon 1/r. Molecules are held together by Molecular Orbitals, which result from the combination of atomic orbitals. Therefore, in order to model a molecule computationally, we must combine Slater Type Orbitals. Unfortunately, Slater Type Orbitals are computationally demanding to manipulate and combine. In order to get around this problem and to increase computational speed, each STO can be approximated by a number of Gaussian Type Orbitals (GTOs). Gaussians are dependant upon 1/r2 and thus by themselves do not accurately describe an atomic orbital. However, a combination of a number of Gaussians is capable of describing an atomic orbital accurately. Gaussians are much easier to manipulate computationally, as the combination of two Gaussians merely gives another Gaussian. These functions are illustrated in '''Figure 1'''.

[[Image:Bc608_FUNCTIONS.png|thumb|centre|300px|'''''Figure 1:''''' ''The functions involved in approximating the atomic orbitals<ref name="STOGTO">P. Hunt, ''Inorganic Computational Chemistry Year 4 lecture materials'' 2011, Imperial College London</ref>'']]

In order to make a computational description of a system more accurate, more STOs can be used to approximate each AO and more GTOs can be used to describe each STO. However, as the number of functions involved increases, so does the computational cost of the calculations. Therefore the number of functions used to describe a system must be carefully considered and a balance between computational cost and accuracy achieved.

When choosing a basis set, the first important factor to be considered is the number of STOs that will be used to approximate each AO. As more STOs are used to describe each AO, the accuracy of the calculations to reality is improved, but the computational cost is increased. Rather than increasing the number of STOs equally for all AOs in the system therefore, it makes more sense to be selective about which AOs are described more accurately. Most chemistry, reactivity and bonding occurs via valence orbitals, therefore a more accurate description of valence AOs will have more of an effect on accuracy than increasing the description of core AOs. Advanced basis sets, often utilise multiple STOs to describe each valence orbital, whereas core AOs are described by only one STO.

The 3-21G basis set used in the optimisation of BH3 is a minimal basis set, that is it uses the minimum number of STOs to describe each occupied AO of the system. Thus one STO is used to describe each occupied AO. In the case of BH3 this corresponds to the hydrogen 1s, the boron 1s, the boron 2s and the boron 2p atomic orbitals.

The next important factor is the number of GTOs that will be used to describe each STO. As before, increasing the number of GTOs increases the approximation of the STO, but also increases the computational cost. The 3-21G basis set is a "split valence" basis set, in which the method of STO approximation is different for core and valence orbitals.

For core orbitals, 3 GTOs are combined to describe each STO. The STO is approximated by a function of the form: f(x) = aGTO1 + bGTO2 + cGTO3, where a,b and c are weighting coefficients.

For valence orbitals, 3 GTOs are also used to describe each STO, but the 3 GTOs are contracted in a pattern of 2:1. This means, two of the GTOs share the same weighting coefficient. The STO is therefore approximated by a function of the form: f(x) = d (GTO1 + GTO2) + eGTO3, where d and e are weighing coefficients.

The justification for treating core orbitals more accurately is that the energy of these orbitals will dominate the total energy of the system and thus their accurate calculation is important.

One would expect the 3-21G basis set to therefore provide an ok approximation of the total energy of the system, but to reflect reactivity poorly, due to the contracted GTOs used in the approximation of the valence STOs. Contraction of valence GTOs is performed to increase computational speed.

Now the basis set has been understood, it is important to ask if it will accurately model BH3. The system to be modelled is very simple and highly symmetric, therefore it is likely that the simple description will be sufficient.

The structure of BH3 was optimised using DFT calculations using B3LYP functional and a 3-21G basis set. A Jmol of the optimised structure is shown to the right. The log file can be found here: {{DOI|10042/to-7581}} The summary of the calculation is shown in '''Figure 2'''.

[[Image:BH3_BUTTON.png|thumb|right|100px| ''The structure of BH3 optimised using DFT/B3LYP using a 3-21G basis set''
<jmol>
<jmolAppletButton>
<uploadedFileContents>BH3_D3H_Point_group_Constraint_SCAN_GEO.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>]]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="2" |'''BH3 Optimisation'''
|-
| '''File type'''||.log
|-
| '''Calculation Type'''||FOPT
|-
| '''Calculation Method'''|| RB3LYP
|-
| '''Basis Set'''||3-21G
|-
|'''E(RB+HF-LYP)'''||-26.462 Hatrees +/- 3.808x10-3
|-
| '''RMS Gradient Norm'''||0.000 a.u. +/- 3.808x10-3
|-
| '''Imaginary Frequencies''' || -
|-
| '''Dipole moment'''|| 0.00 debye +/- 0.005
|-
| '''Point group'''|| D3H
|-
| '''Job time'''|| 16.0 seconds
|} '''''Figure 2:''''' ''A table to show the summary of the optimisation of BH3

The log file was checked in order to assume that the optimisation had converged. The relevant section in the log file is shown in '''Figure 3'''. It was seen that all parameters had converged. This is a necessary step in order to ensure the job has completed properly and has not failed.

Item Value Threshold Converged?
Maximum Force 0.000090 0.000450 YES
RMS Force 0.000059 0.000300 YES
Maximum Displacement 0.000352 0.001800 YES
RMS Displacement 0.000230 0.001200 YES
Predicted change in Energy=-4.580970D-08
Optimization completed.
-- Stationary point found.
'''''Figure 3:''''' ''The section of the output file which showed the calculation had converged''

The optimisation procedure proceeded through 5 iterations, the structure of each iteration is shown in '''Figure 4'''.

[[Image:Optimisation_Structures_1_to_5.png|thumb|centre|600px|'''''Figure 4:''''' ''Structures 1 to 5 explored in the optimisation procedure'']]

In order to understand the optimisation procedure, it's imperative to first recall the Born Oppenheimer approximation.

The Born Oppenheimer approximation states that electrons move so very much faster than the relatively slow nuclei. The electrons can therefore change their positions instantaneously to respond to a change in nuclei position and therefore the momenta of nuclei does not affect the wavefunction of the electrons i.e. from the perspective of the electrons, the nuclei are stationary. It is important to note that the wavefunction of the electrons still depends upon the positions of the nuclei as there is a Coulombic attraction between the electrons and the nuclei. However, it does not depend upon the nuclei's momenta.

During the optimisation procedure, the energy of the system is calculated by solving the Schrodinger equation under the Born Oppenheimer approximation for a given set of nuclear positions. If all the nuclear positions are represented by R, then the energy calculated by solving the Schrodinger equation at these nuclear positions is E(R). The forces of the interactions between electrons and nuclei are then calculated at R. The nuclear positions are then altered i.e. R is changed to R'. The energy E(R') is then calculated and the forces calculated at R'. For every iteration, the absolute value of the rate of change of E(R) with R is calculated and this is the Root Mean Squared Gradient.

The 'optimised' structure is defined as the structure of lowest energy and is therefore a minima on the global potential energy surface. The process of reaching this minima from a starting point is termed "optimisation". As minima are a type of stationary point, the root mean squared gradient at the optimised structure is 0. The procedure of energy calculation, gradient calculation and nuclear position adjustment is repeated iteratively until a stationary point is reached.

The optimisation procedure is shown graphically in '''Figure 5'''.

[[Image:TheIterationProcedure.png|thumb|centre|300px|'''''Figure 5:''''' ''A cartoon to show the iterative optimisation procedure. Blue dots show a successful convergence to the most stable structure over 3 iterations. Dotted red lines show the gradient of the tangent line to the E(R) curve. The yellow dot shows a maxima that may be converged to if a different starting structure is chosen. The green dot shows a false minimum that may be converged to if a different starting structure is chosen.'']]

The optimisation procedure for BH3 is shown in '''Figure 6'''.

[[Image:BH3_Opt_Graphs_reformatted.png|thumb|centre|900px|'''''Figure 6:''''' ''Graphs to show to optimisation procedure for BH3'']]

Whilst a stationary point has been found, it is not certain that the job has converged to a minimum and even if it has, if the right minimum has been converged to! In order to ensure this, analysis of the vibrational frequencies has to be performed.

If we are at a minima, then any movement of atoms and therefore electrons relative to others(e.g. a stretch of a bond) will result in a higher energy state and thus energy will have to be invested to make the movement happen. However, if we were at a maxima, a specific change in R will result in a decrease in E(R) i.e. energy is released when the movement happens.

This provides a means of distinguishing between convergence to a maxima or a minima. Minima will have entirely positive vibrational frequencies. Maxima will possess one negative vibrational frequency. If more than one negative vibrational frequency is present then a stationary point has not been found.

====Vibrational analysis====

=====Vibrational analysis to confirm a minima and comparison of measured parameters to literature=====

The vibrational frequencies of the optimised structure of BH3 were calculated using DFT-B3LYP and a 3-21G basis set.

It's important to note that the same calculation type and basis set is necessary. The 'optimised structure' is only a minima on the Potential Energy Surface calculated using DFT-B3LYP and a 3-21G basis set. The 'optimised structure' is not necessarily a minima on the PES calculated by other methods or other basis sets. If the calculation were performed using a different calculation type, method or basis set, then it is highly likely that one or more negative frequencies would result. This illustrates an important point: results obtained from calculations are accurate only to the level of theory and basis set used. Different methods and basis sets will give different results!

A summary of the frequency calculation is shown below in '''Figure 7'''. {{DOI|10042/to-7486}}

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="2" |'''Vibrational analysis of BH3'''
|-
| '''File type'''||.log
|-
| '''Calculation Type'''||FREQ
|-
| '''Calculation Method'''|| RB3LYP
|-
| '''Basis Set'''||3-21G
|-
|'''E(RB+HF-LYP)'''||-26.462 Hatree +/- 3.808x10-3
|-
| '''RMS Gradient Norm'''||0.000 Hatree +/- 3.808x10-3
|-
| '''Imaginary Frequencies''' || 0
|-
| '''Dipole moment'''|| 0.00 debye +/- 0.005
|-
| '''Point group'''|| D3H
|-
| '''Job time'''|| 17.4 seconds
|} '''''Figure 7:''''' ''A table to show the summary of the calculation''

The total energy of the system was found to be the same as the energy recorded for the optimisation. This confirmed the optimisation had been performed on the correct file.

The frequencies found to be result of translation of the whole molecule without bond distortion were found to be: 0, 0, 0, 37, 38 and 38 cm-1. These frequencies should be close to zero - the more accurate the method and basis set employed the closer to zero these values will be.

The lack of proximity to zero in this case can be attributed to the limited basis set utilised in this calculation, as discussed previously. However, the largest of these frequencies 38 cm-1 is still 2 orders of magnitude smaller than the lowest frequency vibration (1146 cm-1). For the purposes of this experiment, this will suffice. The results from the frequency analysis are shown in '''Figure 8'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''|| '''Form of the vibration'''|| '''Description'''|| '''Frequency (error in frequency) /cm-1'''|| '''Intensity''' || '''Symmetry within the D3H point group'''
|-
| 1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>BH3_FREQ_CALC_OUT_FILE.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Out of plane wagging (umbrella motion)</text>
</jmolAppletButton>
</jmol>
|| The 3 hydrogens move together in the z direction, with the boron moving in the opposite direction at all times so that the Centre of Mass of the molecule does not change over the vibration|| 1146 (± 115) || 93|| A2<nowiki>''</nowiki>
|-
| 2 || <jmol>
<jmolAppletButton>
<uploadedFileContents>BH3_FREQ_CALC_OUT_FILE.out</uploadedFileContents>
<script>frame 4; vectors 4; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>In plane scissoring (bending motion) </text>
</jmolAppletButton>
</jmol>
|| One H-B-H angle is contracted, whilst the two other H-B-H angles are expanded. The two hydrogens moving down is counteracted by the remaining Boron and hydrogen moving upwards so that the centre of mass does not change over the vibration|| 1205 (± 121)|| 12|| E'
|-
| 3 ||<jmol>
<jmolAppletButton>
<uploadedFileContents>BH3_FREQ_CALC_OUT_FILE.out</uploadedFileContents>
<script>frame 5; vectors 4; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>In plane rocking (bending motion)</text>
</jmolAppletButton>
</jmol>
|| One H-B-H angle is held constant, whilst another contracts and the third is expanded. The large movement of the hydrogen moving to change the two H-B-H angles is counteracted by smaller movements of the remaining atoms so that the Centre of Mass does not change over the vibration || 1205 (± 121)|| 12 || E'
|-
| 4 || <jmol>
<jmolAppletButton>
<uploadedFileContents>BH3_FREQ_CALC_OUT_FILE.out</uploadedFileContents>
<script>frame 6; vectors 4; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Symmetric Stretching</text>
</jmolAppletButton>
</jmol>
|| All B-H bonds stretch symmetrically at the same time. The central Boron does not move throughout the vibration. The symmetry of the vibration means that the Centre of Mass does not change over the vibration.|| 2592 (± 259)|| 0 || A1'
|-
| 5 ||<jmol>
<jmolAppletButton>
<uploadedFileContents>BH3_FREQ_CALC_OUT_FILE.out</uploadedFileContents>
<script>frame 7; vectors 4; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Asymmetric stretching</text>
</jmolAppletButton>
</jmol>|| One B-H bond has a constant length, the remaining two lengthen and shorten asymmetrically. The hydrogen not involved in stretching and the Boron are moved to counteract the movements of hydrogens so that the centre of mass does not change over the vibration.|| 2730 (± 273) || 104 || E'
|-
| 6 ||<jmol>
<jmolAppletButton>
<uploadedFileContents>BH3_FREQ_CALC_OUT_FILE.out</uploadedFileContents>
<script>frame 8; vectors 4; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Asymmetric stretching</text>
</jmolAppletButton>
</jmol>|| Two B-H bonds stretch symmetrically, whereas the third stretches asymmetrically with respect to the other two. When the two B-H bonds lengthen, the third will shorten and the Boron will travel upwards so that there is no change in the centre of mass over the vibration.|| 2730 (± 273)|| 104 || E'
|} '''''Figure 8:''''' ''A table to show the summary of the calculation''

In the vibrations, not only are the hydrogen atoms moving, but the
important Although it is nice to have animations of the vibrations, the purpose of this part is to help you think about the atomic movements and this is achieved by you drawing out and describing the different motions, from an understanding point of view just making a movie of the vibration doesn't do this, so a maximum of ONE animation please! (lots of animations also cloggs up your wiki and makes it slow)

The lack of any negative stretching frequencies shows the optimised structure is in fact an energy minimum, but does not show that it is the global minimum. In order to be certain the structure has been correctly optimised, a comparison must be made to experimental data. In this case, a comparison of bond lengths and angles is possible.

The B-H bond distance was measured to be 1.19 Å ± 0.005 and the H-B-H angle measured to be 120° ± 0.05. Reported values for B-H bond lengths in a similar structure, BH4-, are of the order of 1.21 Å <ref name="BH3Parameters">M.R. Hartman, J.J. Rush, T.J. Udovic, R.C. Bowman and S.J. Hwang, ''Jounral of Solid State Chemistry'' 2007, '''180''', 1298-1305 {{DOI|10.1016/j.jssc.2007.01.031}}</ref>. Thus the measured bond length does not seem unrealistic and it is likely that the structure has been optimised successfully.

Most importantly, the optimised structure makes chemical sense - one would expect BH3 to adopt a trigonal planar structure.

=====Vibrational analysis to calculate the IR spectrum of BH3=====

The results of the Vibrational Analysis are shown in '''Figure 8'''. Symmetry of each vibration was determined using a representation table. This process is illustrated in '''Figure 9''' for the first vibration of A2<nowiki>''</nowiki> symmetry. Each symmetry operation of the point group is performed upon the vibration. If the vibration remains unchanged under a symmetry operation, the element of the vibration under that symmetry operation is 1. If the vibration is reversed, the element of the vibration under that symmetry operation is -1.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_SymmetryElementsofBH3.png|centre|400px|]] || [[Image:Bc608_SymmetryElementsofBH3_2.png|centre|200px|]]

|}

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''D3h|| '''E'''|| '''2C3'''|| '''3C2''' || '''σh''' || '''2S3''' || '''3σv'''
|-
| '''Γ''' [[Image:Bc608_UnderE.png|centre|60px]] || 1 || 1 || -1 || -1 || -1 || 1
|} '''''Figure 9''''' : ''The representation table for the A2<nowiki>''</nowiki> stretch

This procedure was performed for all the above vibrations. The IR spectrum was also calculated and is shown in '''Figure 10'''.

[[Image:IR_SPECTRUM_OF_BH3.png|thumb|centre|800px|'''''Figure 10:''''' ''The calculated IR spectrum of BH3'']]

It can be seen that there are only 3 peaks observed in the IR spectrum of BH3, however we have calculated 6 vibrations!

In order to see a peak in an IR spectrum, light must be absorbed in the IR frequency range. In order for an absorption to occur, there must be a non-zero transition dipole moment. The dependency of transition dipole moment upon change in molecular dipole over the vibration is shown in the following equation:

[[Image:Bc608_TRANSITIONDIPOLEMOMENT.png|centre|200px|]]

Where:
'''μα''' is the transition dipole moment

'''(δμ/δQk)0''' is the change in molecular dipole moment over the vibrational mode in question '''Qk'''

'''φi''' is the vibrational wavefunction at equilibrium

'''φj''' is the vibrational wavefunction at vibrational displacement j

It can therefore be seen that vibrational modes that do not give a change in molecular dipole moment over the vibration will have transition dipole moments equal to zero and therefore these vibrational modes are not IR active.

For mode #4, there is no change in dipole moment over the vibration because the stretching is totally symmetric, one would therefore expect this band to be IR inactive. This leaves 5 vibrational modes that display a change in dipole moment over the vibration.

It is now important to realise that vibrations #2 and #3 are degenerate. They are of the same energy and so occur at the same frequency - this is denoted by their e' label. The peak at 1205 cm-1 is the result of not one, but two vibrations of the same frequency. Vibrations #5 and #6 are also degenerate and both occur at 2730 cm-1. The third band occurs at 2592 cm-1 and is singly degenerate. Thus 3 peaks are observed in the IR spectrum, corresponding to 5 vibrations: 2 IR active degenerate pairs of vibrations and a 5th IR active vibration. The 6th vibration is totally symmetric and IR inactive.

====Population Analysis: Molecular Orbitals of BH3, a comparison of qualitative and quantitative molecular orbital methods====

The energy of the optimised structure of BH3 was calculated using DFT-B3LYP, a 3-21G basis set and full NBO and population analysis. {{DOI|10042/to-7487}}

=====Qualitative Molecular Orbital Diagrams=====

A qualitative molecular orbital diagram was generated and can be seen in '''Figure 12'''. The fragment orbitals of H3 are constructed using a representation table (as in '''Figure 9'''), the reduction formula and the projection formula. Hydrogen and Boron are of similar electropositivity and thus of similar energy. The fragment orbitals of H3 all display bonding character, thus the a1' has been placed below the 1s orbital of Boron to reflect it being of lower energy. The e' levels of H3 have been estimated to be of comparable energy to the boron p orbitals. The orbitals are then mixed according to their symmetry labels. Only orbitals of the same symmetry can interact and each orbital only interacts once. Secondary orbital mixing occurs between molecular orbitals of the same symmetry, but it was judged to be insignificant in the case of BH3 as there are no orbitals of the same symmetry that are close in energy.

The relative ordering of the 3a' and the 2e' molecular orbitals is hard to evaluate qualitatively. The relative energies of molecular orbitals are given by the Klopman-Salem equation, which can be expressed in the following simplified form:

[[Image:Klopman-salem.png|centre|200px]]

'''S2''' is the overlap integral
'''ESTAB''' is the stabilisation energy
'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

By examining how the S2 and ΔEi of the Klopman-Salem equation vary for each overlap of fragment orbital, one determines the stabilisation/destabilisation of the resulting molecular orbitals relative to the initial fragment orbitals. By combining the stabilisation energy and the unperturbed fragment orbital energy, one can estimate the energy of a molecular orbital.

This analysis is fine for the majority of the orbitals of BH3, however a problem arises when considering the ordering of the 3a1' and 2e' MOs. Applying the above analysis:

'''Overlap:''' S orbitals overlap more effectively with other s orbitals than they do with p orbitals, so one would expect a larger S2 term for the 3a' molecular orbital than for the 2e' molecular orbital.

'''Interaction Energy:''' This is very hard to judge.

'''Fragment orbital energies:''' The a1' orbital on H3 is lower in energy than the e' orbitals. The 2s orbital of the Boron atom is lower in energy than the 2px and 2py orbitals. Thus the unperturbed fragment orbital energy is lower for the 3a1' MO than for the 2e' MO.

There are two opposite acting effects and a third that cannot be determined. Qualitative treatment gives us no indication which effect is larger. In order to determine the ordering of energy levels a quantitative treatment is required.

=====Quantitative Molecular Orbital Diagrams=====

The MOs from the population analysis were visualised. It can be seen from '''Figure 11''' that the 3a' MO is in fact higher in energy than the 2e' MO at this level of theory. As previously mentioned, results can change dependent upon the basis set and calculation type used.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Orbital'''||'''Orbital Energies ± 3.808x10-3 /Hatree'''
|-
| 1a1'|| -6.726
|-
| 2a'1 || -0.520
|-
| 1e' || -0.358
|-
| 1a''2 || -0.074
|-
| 2e' || 0.192
|-
| 3a'1|| 0.199
|} '''''Figure 11:''''' ''A table to show the relative orbital energies of the BH3 molecular orbitals

[[Image:Bc608_CalculatedMODiagram.png |thumb|centre|1000px|'''Figure 12:''' ''A qualitative molecular orbital diagram with calculated molecular orbitals illustrated alongside for comparison]]

On the whole, the calculated molecular orbitals are in excellent agreement with the qualitative LCAO molecular orbitals. The agreement between molecular orbitals is less good for unfilled molecular orbitals however, as can be seen by the right hand 2e' orbital (or perhaps this is just difficult to visualise.) In the mini project, most of the orbitals above the LUMO were very difficult to discern at least, so this statement is qualified by observations there if not here.

Qualitative molecular orbital theory is useful and accurate to an extent. It provides a quick means of understanding reactivity of compounds, for instance Natural Bond Orbitals can be used qualitatively to understand relatively complex processes, such as the Eschenmoser fragmentation.<ref name="Echernmoser">A. Eschenmoser, D. Felix and G. Ohloff , ''Helvetica Chimica Acta'' 1967, '''50''', 708-713 {{DOI|doi:10.1002/hlca.19670500232}}</ref>

Whilst the ordering of energy of the 2e' and 3a1' molecular orbitals is difficult to determine qualitatively, in order to understand the reactivity of BH3 it doesn't much matter which orbital is lower in energy. The energy levels of the HOMO and LUMO are unambiguous in this case and it is these orbitals that will dominate reactivity. Computational MO theory is more useful for complex systems, where deriving the molecular orbitals "by hand" is more difficult.

====Natural Bond Orbital Analysis of BH3: Charge distribution====

NBO analysis on the optimised structure of BH3 was performed in an earlier calculation. {{DOI|10042/to-7487}}

NBO analysis takes a delocalised MO picture and returns a 2 electron-2 centre picture of bonding.

The log file was opened and the summary of Natural Population Analysis found. The summary shows the natural atomic charges and the population of core orbitals, valence orbitals and Rydberg orbitals (i.e. very diffuse orbitals that have principal quantum numbers higher than the atoms' valence orbitals). For instance the 1.99902 in the Core column of Boron represents the two electrons that are in the 1s atomic orbital of the Boron atom. This is shown in '''Figure 13'''.

Natural Population
Natural -----------------------------------------------
Atom No Charge Core Valence Rydberg Total
-----------------------------------------------------------------------
B 1 0.33679 1.99902 2.66419 0.00000 4.66321
H 2 -0.11226 0.00000 1.11193 0.00033 1.11226
H 3 -0.11226 0.00000 1.11193 0.00033 1.11226
H 4 -0.11226 0.00000 1.11193 0.00033 1.11226
=======================================================================
* Total * 0.00000 1.99902 6.00000 0.00099 8.00000
'''Figure 13:''' ''The summary of Natural Population Analysis''

Gaussview can interpret the natural charges to give a more visual picture of the natural charges in the molecule. These are shown in '''Figure 14'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_BH3_MO_CHARGEDISTRIBUTION_NBO.png|thumb|centre|200px]]
||
[[Image:Bc608_BH3_MO_CHARGEDISTRIBUTION_NBO_NUMBERS.png|thumb|centre|250px]]
|} '''Figure 14:''' ''The charge distribution in BH3 as calculated by NBO analysis. Red denotes negative charge whereas green denotes positive charge. Numbers represent the NBO charges of the atoms.

The Boron is BH3 is shown to be highly electron deficient, as would be expected based upon the Lewis structure of the compound.

Examining the log file further, one can determine the bonding in the compound in terms of hybridisation.

(Occupancy) Bond orbital/ Coefficients/ Hybrids
---------------------------------------------------------------------------------
1. (1.99884) BD ( 1) B 1 - H 2
( 44.40%) 0.6663* B 1 s( 33.33%)p 2.00( 66.67%)
0.0000 0.5774 0.0000 0.0000 0.0000
0.8165 0.0000 0.0000 0.0000
( 55.60%) 0.7457* H 2 s(100.00%)
1.0000 0.0000
2. (1.99884) BD ( 1) B 1 - H 3
( 44.40%) 0.6663* B 1 s( 33.33%)p 2.00( 66.67%)
0.0000 0.5774 0.0000 0.7071 0.0000
-0.4082 0.0000 0.0000 0.0000
( 55.60%) 0.7457* H 3 s(100.00%)
1.0000 0.0000
3. (1.99884) BD ( 1) B 1 - H 4
( 44.40%) 0.6663* B 1 s( 33.33%)p 2.00( 66.67%)
0.0000 0.5774 0.0000 -0.7071 0.0000
-0.4082 0.0000 0.0000 0.0000
( 55.60%) 0.7457* H 4 s(100.00%)
1.0000 0.0000
4. (1.99902) CR ( 1) B 1 s(100.00%)
1.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
5. (0.00000) LP*( 1) B 1 s(100.00%)
6. (0.00000) RY*( 1) B 1 s( 0.00%)p 1.00(100.00%)
7. (0.00000) RY*( 2) B 1 s( 0.00%)p 1.00(100.00%)
8. (0.00000) RY*( 3) B 1 s( 0.00%)p 1.00(100.00%)
9. (0.00000) RY*( 4) B 1 s( 0.00%)p 1.00(100.00%

'''Figure 15:''' ''The NBOs in BH3''

The first three Natural Bond orbitals are between the Boron and a Hydrogen, 44.40% of each bond is contributed to by Boron orbitals that are sp2 hybridised (33.33% s character, 66.67% p character). 55.60% of each bond is contributed to by Hydrogen orbitals that are exclusively s character - as one would expect.

The 4th Natural Bond orbital is a core orbital centred on Boron, which is 100% s character. This will be the occupied 1s orbital of Boron, which does not participate in bonding.

The 5th Natural Bond orbital is interesting. One would expect the 5th orbital to be an unhybridised empty p orbital of boron, however the logfile has reported an empty orbital of 100% s character. The electronic structure of Boron is 1s22s22p2. The third p orbital of Boron must have been included in the minimal basis as one cannot determine which two of the three p orbitals the 2 p electrons reside in, therefore it is not entirely clear where it has gone.

The example in the lab instructions does indeed give the 5th NBO as an empty boron p orbital as expected. However, it is important to note that many of the example calculations were performed with a 6-31G basis set. It is therefore likely that the aphysical result seen using a 3-21G basis set is due to a limitation of applying this basis set to this calculation. It must be stressed that the limitation is '''not''' the result of a lack of inclusion of this third p orbital in the basis set, as it must have been included to describe a Boron atom. By examining the log file further, the empty Boron p orbital seems to being treated as a Rydberg orbital in higher NBOs. The cause of this is unknown.

The "Second Order Perturbation Theory Analysis of Fock Matrix in NBO Basis" section outlines mixing of MOs, however it does not show much information for BH3 as no significant second order mixing takes place.

Second Order Perturbation Theory Analysis of Fock Matrix in NBO Basis
Threshold for printing: 0.50 kcal/mol
E(2) E(j)-E(i) F(i,j)
Donor NBO (i) Acceptor NBO (j) kcal/mol a.u. a.u.
===================================================================================================
within unit 1
4. CR ( 1) B 1 / 10. RY*( 1) H 2 1.54 7.54 0.096
4. CR ( 1) B 1 / 11. RY*( 1) H 3 1.54 7.54 0.096
4. CR ( 1) B 1 / 12. RY*( 1) H 4 1.54 7.54 0.096

'''Figure 16:''' ''Molecular orbital mixing in BH3''

Only values in the E(2) column greater than 20kcal/mol are deemed to be significant. It would seem there is an insignificant interaction here between the Boron 1s orbital and the Hydrogen's empty Rydberg orbitals, which is stabilising the Boron 1s orbitals slightly.

The last section gives a summary of the occupancy of the NBOs of BH3.

Natural Bond Orbitals (Summary):

Principal Delocalizations
NBO Occupancy Energy (geminal,vicinal,remote)
====================================================================================
Molecular unit 1 (H3B)
1. BD ( 1) B 1 - H 2 1.99884 -0.43956
2. BD ( 1) B 1 - H 3 1.99884 -0.43956
3. BD ( 1) B 1 - H 4 1.99884 -0.43956
4. CR ( 1) B 1 1.99902 -6.63856 10(v),11(v),12(v)
5. LP*( 1) B 1 0.00000 0.67133
6. RY*( 1) B 1 0.00000 0.37047
7. RY*( 2) B 1 0.00000 0.37047
8. RY*( 3) B 1 0.00000 -0.04349
9. RY*( 4) B 1 0.00000 0.43436
10. RY*( 1) H 2 0.00033 0.89768
11. RY*( 1) H 3 0.00033 0.89768
12. RY*( 1) H 4 0.00033 0.89768
13. BD*( 1) B 1 - H 2 0.00116 0.42928
14. BD*( 1) B 1 - H 3 0.00116 0.42928
15. BD*( 1) B 1 - H 4 0.00116 0.42928
-------------------------------
Total Lewis 7.99554 ( 99.9442%)
Valence non-Lewis 0.00348 ( 0.0435%)
Rydberg non-Lewis 0.00099 ( 0.0123%)
-------------------------------
Total unit 1 8.00000 (100.0000%)
Charge unit 1 0.00000

'''Figure 16:''' ''A summary of the NBO analysis in BH3''

It can be seen that there are 2 electrons in each of the B-H bonds, 2 electrons in the Boron 1s and an empty orbital on Boron which is being treated as s character, but we know to be p character.

===TlBr3===

====Pseudo potentials====

TlBr3 is a compound that has 151 electrons. At the level of quantum mechanics being utilised in these calculations, computing the wavefunction for each of these orbitals would be computationally very expensive. Furthermore, the size of the atoms causes them to exhibit relativistic effects, which are not recovered easily by using the standard Schrodinger equation. In order to get around this problem, a pseudo potential is used. A pseudo potential is a potential used to model the core orbitals of an atom. This approximation is justified by the knowledge that core orbitals of atoms are not heavily involved in chemical bonding, which in the large part can be attributed to valence orbitals.

Furthermore, as a group is descended, valence orbitals become more diffuse and polarizable. Both of the atoms in this calculation are beyond the first row of the periodic table and thus these effects will be significant. The 3-21G basis set previously used therefore needs to be improved upon because the system is much more complex than was previously the case.

A LANL2DZ basis set is used to optimise TlBr3. This basis set uses the Los Alamos Effective Core Potential for core orbitals and double zeta basis functions for the valence atomic orbitals. "Double zeta basis functions" means that 2 Slater Type Orbitals are used to model each atomic orbital, which gives an improved description of the valence orbitals. This approach of improved description of the valence orbitals with multiple STOs, with a less accurate description of core orbitals was discussed in the '''Optimisation of BH3''' section of this report.

====Optimisation====

[[Image:TlBr3.png|thumb|right|100px| ''The structure of TlBr3 optimised using DFT/B3LYP using a LanL2DZ basis set''
<jmol>
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<uploadedFileContents>Bc608_TlBr3_molfile.mol</uploadedFileContents>
<text>Jmol</text>
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The structure of TlBr3 was optimised using DFT-B3LYP and a LANL2DZ basis set. The optimised structure was constrained to be in the D3h point group to avoid false minima. The optimised structure is shown to the right. The logfile for the optimisation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/4/4a/Bc608_TLBR3_GEO_OPT.LOG

A summary of the calculation is shown in '''Figure 17'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="2" |'''Optimisation of TlBr3'''
|-
| '''File type'''||.log
|-
| '''Calculation Type'''||FOPT
|-
| '''Calculation Method'''|| RB3LYP
|-
| '''Basis Set'''||LANL2DZ
|-
|'''E(RB+HF-LYP)'''||-91.218 Hatree +/- 3.808x10-3
|-
| '''RMS Gradient Norm'''||0.000 Hatree +/- 3.808x10-3
|-
| '''Imaginary Frequencies''' || 0
|-
| '''Dipole moment'''|| 0.00 debye +/- 0.005
|-
| '''Point group'''|| D3H
|-
| '''Job time'''|| 26.0 seconds
|} '''''Figure 17:''''' ''A table to show the summary of the calculation''

====Frequency Analysis to confirm a minimum and comparison of parameters to literature values====

As for BH3, a confirmation that a minimum has been reached is needed. Frequency analysis was performed on the optimised structure of TlBr3 using DFT-B3LYP and a LANL2DZ basis set. A summary of the calculation is shown in '''Figure 18'''. The log file of the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/b/b5/BC608_TLBR3_FREQ.LOG A discussion of why the same method and basis set must be used was provided in the optimisation of BH3 earlier in this report.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="2" |'''Frequency Analysis of TlBr3'''
|-
| '''File type'''||.log
|-
| '''Calculation Type'''||FREQ
|-
| '''Calculation Method'''|| RB3LYP
|-
| '''Basis Set'''||LANL2DZ
|-
|'''E(RB+HF-LYP)'''||-91.218 Hatree +/- 3.808x10-3
|-
| '''RMS Gradient Norm'''||0.000 Hatree +/- 3.808x10-3
|-
| '''Imaginary Frequencies''' || 0
|-
| '''Dipole moment'''|| 0.00 debye +/- 0.005
|-
| '''Point group'''|| D3H
|-
| '''Job time'''|| 22.0 seconds
|} '''''Figure 17:''''' ''A table to show the summary of the calculation''

The low frequencies of TlBr3 were found to be -3, 0, 0, 0 and 4 cm-1.

The lowest 'real' mode of TlBr3 was found to be 46cm-1. The lowest low frequency was found to be an order of magnitude smaller than the lowest 'real' frequency and so the level of accuracy of the calculation is acceptable. The vibrational frequencies of TlBr3 are shown in '''Figure 18'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''|| '''Frequency (error in frequency) /cm-1'''|| '''Intensity'''
|-
| 1 || 46 (± 5) || 4
|-
| 2 || 46 (± 5)|| 4
|-
| 3 || 52 (± 5)|| 6
|-
| 4 || 165 (± 17)|| 0
|-
| 5 || 211 (± 21) || 25
|-
| 6 || 211 (± 21) || 25
|} '''''Figure 18:''''' ''A table to show the frequencies of the vibrations of TlBr3

All the vibrations of TlBr3 were found to be positive. The absence of negative frequencies shows that a minimum has been reached.

Now a comparison to literature is required to check that the optimised structure is reasonable. The Tl-Br bond length in TlBr3 was measured to be 2.65Å ± 0.005 and the Br-Tl-Br bond angle measured to be 120.0° ± 0.05. The length of the Tl-Br bond was measured to be 2.512Å experimentally<ref name="TlBr3">J. Glaser and G. Johansson, ''Acta Chemica Scandinavica'' 1982, '''36a''', 125-135 {{DOI|10.1021/jp0124802}}</ref>, so this value does not seem unrealistic. The bond angle of 120° makes intuitive sense. The structure of TlBr3 has therefore been successfully optimised using pseudo potentials.

====What is a bond?====

In some structures, Gaussview does not draw in bonds where we expect them to be. This is seen for the Molybdenum complex discussed after this section, where the P-Cl bonds are not drawn in. It is important to remember that Gaussview is merely a graphical interface that is used to set up calculations and view output files that Gaussian performs and creates. When interpreting output files to provide a visual image, Gaussview uses the proximity of atoms to decide the bond order between atoms. When Gaussview fails to display a bond, it is because Gaussview decides by comparison to standard bond lengths, the atoms are far apart for a bond to exist. Gaussview does '''not''' make any attempt to consider electron density between atoms when deciding whether a bond is present.

This is obviously a major limitation of Gaussview. A bond is a region of high electron density between two or more atoms that results in attractive forces between the nuclei involved. A bond between atoms cannot exist without shared electron density between the atoms. Thus it is clear that a mere distance relationship between nuclei is insufficient to describe a "bond" and electron density between involved atoms must be considered.

Gaussview may not draw the bonds in, but they are still there.

===Cis and trans isomerisation of Mo(CO)4L2===

In the second year synthesis lab, cis- and trans-Mo(CO)4PPh3 was prepared. 4 carbonyl absorption bands were observed from the cis complex, whereas only one band was observed from the trans complex.

In order to understand this observation, it would be desirable to compute the vibrations of the molecule. Unfortunately, performing calculations on the complexes directly will be too computationally expensive due to the bulky triphenylphosphines. However, a model system can be utilised, where computationally less demanding Cl atoms replace the phenyl groups. Chlorines could be expected to have similar electronic contributions to bonding as phenyl groups and like phenyl groups they are also sterically large. The cis and trans isomers of Mo(CO)4PCl3 were optimised with a successive optimisation procedure outlined below.

====Initial loose optimisation====

In the exercise, it was stated that Dr Hunt had searched over the rotational profile in order to find a good starting point for the optimisation. It is important to be selective with the starting point of the calculation in order to avoid false minima. As previously discussed in '''Figure 3''', false minima are defined as local minima that are not the overall most stable structure of the molecule. Calculations may converge to these local minima as shown in '''Figure 19'''.

[[Image:Explainationoflocalandglobalminima.png|thumb|centre|400px| '''Figure 19:''' ''A cartoon to show convergence to the global minima (red) and a local minima (blue)']]

The starting geometries for the optimisations are shown in '''Figure 20'''.

[[Image:StartingGeometries.png|thumb|centre|400px| '''Figure 20:''' ''The starting geometries of the cis and trans complexes.'']]

From these starting geometries, the DFT-B3LYP method was utilised using a LANL2MB basis set and loose optimisation conditions in order to get a rough optimisation so that the subsequent calculation would be more rapid. LANL2MB utilises a pseudo potential for core orbitals, whilst a minimal basis (1 STO for each AO) is used for the valence orbitals. Considering the complexity of the structure, this is not sufficient to model the structure, but it is a good starting point to get the geometry "roughly correct" for subsequent optimisations.

The calculation summary is shown in '''Figure 21'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="3" |'''Loose Optimisation of Mo complex'''
|-
| || '''Cis'''|| '''Trans'''
|-
| '''D-Space''' || {{DOI|10042/to-7495}}|| {{DOI|10042/to-7496}}
|-
| '''File type'''||.log||.log
|-
| '''Calculation Type'''||FOPT||FOPT
|-
| '''Calculation Method'''||RB3LYP||RB3LYP
|-
| '''Basis Set'''||LANL2MB||LANL2MB
|-
|'''E(RB+HF-LYP)'''|| -617.525 Hatrees ± 3.808x10-3||-617.522 ± 3.808x10-3
|-
| '''RMS Gradient Norm'''|| 0.000 Hatrees ± 3.808x10-3 ||0.000 ± 3.808x10-3
|-
| '''Dipole moment'''|| 8.62 debye ± 0.005|| 0.31 ± 0.005
|-
| '''Point group'''||C2||CS
|-
| '''Job time'''|| 8 minutes 7.0 seconds|| 5 minutes 46.1 seconds
|} '''''Figure 21:''''' ''A table to show the summary of the loose optimisation calculation

====Double zeta basis set and pseudo potential optimisation====

Using the roughly optimised output from the previous calculation, the geometry was more accurately optimised using the DFT-B3LYP method, a LANL2DZ basis set and ultrafine electronic convergence. As previously discussed, LANL2DZ is a double zeta basis set and uses two Slater Type Orbitals to model each valence atomic orbital. This will give a much better description of the valence atomic orbitals and should describe the bonding in the complex much more accurately.

The job was performed, checked and found to converge. The calculation summary is shown in '''Figure 22'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="3" |'''Double zeta basis set and pseudo potential optimisation'''
|-
| || '''Cis'''|| '''Trans'''
|-
| '''D-Space''' || {{DOI|10042/to-7499}}|| {{DOI|10042/to-7498}}
|-
| '''File type'''||.log||.log
|-
| '''Calculation Type'''||FOPT||FOPT
|-
| '''Calculation Method'''||RB3LYP||RB3LYP
|-
| '''Basis Set'''||LANL2DZ||LANL2DZ
|-
|'''E(RB+HF-LYP)'''|| -623.577 Hatree ± 3.808x10-3||-623.576 Hatree ± 3.808x10-3
|-
| '''RMS Gradient Norm'''|| 0.000 Hatree ± 3.808x10-3||0.000 Hatree ± 3.808x10-3
|-
| '''Dipole moment'''|| 1.3072 debye ± 0.005|| 0.3039 ± 0.005
|-
| '''Point group'''||C2||CS
|-
| '''Job time'''|| 22 minutes 57.8 seconds|| 19 minutes 46.0 seconds
|} '''''Figure 22:''''' ''A table to show the summary of the second optimisation

It is important to note that this calculation implies that the cis complex is more stable than the trans comples by 2.6 KJ mol-1. However, it's important to realise that this difference is small and within the error of the calculation, which is approximately 10 KJ mol-1 One major failing of this calculation is the lack of treatment of phosphorous d orbitals. Phosphorous often displays hypervalency and can utilise low lying d orbitals in order to accomplish this. This complex may show some Phosphorous d character in its bonding and so the phosphorous d orbitals in this compelex cannot be reasonably ignored.

====Including Phosphorous d orbitals====

Phosphorous d orbitals were accounted for by including an extra basis, the following was added to the bottom of the input file.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|(blank line)
|-
|P 0
|-
|D 1 1.0
|-
|0.55 0.100D+01
|-
| ****
|-
| (blank line)
|}

This essentially includes 5 extra atomic orbitals to be calculated corresponding to the unfilled d orbitals on phosphorous. In order to model the 5 new AOs, 10 extra STOs are required (2 for each AO as it is a double zeta basis set), which results in a computationally more demanding calculation.

The cis and trans structures were then reoptimised using DFT-B3LYP, ultrafine electronic convergence and the new adapted basis set. The summaries of the calculations are shown in '''Figure 23'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="3" |'''Double zeta basis set and pseudo potential optimisation'''
|-
| || '''Cis'''|| '''Trans'''
|-
| '''D-Space''' || {{DOI|10042/to-7501}}|| {{DOI|10042/to-7500}}
|-
| '''File type'''||.log||.log
|-
| '''Calculation Type'''||FOPT||FOPT
|-
| '''Calculation Method'''||RB3LYP||RB3LYP
|-
| '''Basis Set'''||LANL2DZ with added d orbitals for phosphorous||LANL2DZ with added d orbitals for phosphorous
|-
|'''E(RB+HF-LYP)'''|| -623.693 Hatree ± 3.808x10-3||-623.694 Hatree ± 3.808x10-3
|-
| '''RMS Gradient Norm'''|| 0.000 Hatrees ± 3.808x10-3 || 0.000 Hatree ± 3.808x10-3
|-
| '''Dipole moment'''|| 0.0731 debye ± 0.005|| 0.2293 ± 0.005
|-
| '''Point group'''||C2||CS
|-
| '''Job time'''|| 19 minutes 3.2 seconds|| 21 minutes 24.7 seconds
|} '''''Figure 23:''''' ''A table to show the summary of the third optimisation

The output files were checked and both jobs found to converge.

====Frequency Analysis to confirm the Minima====

Frequency analysis was performed in order to ensure both isomers were at a minimum. The summary of the calculation is shown in '''Figure 24'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="3" |'''Double zeta basis set and pseudo potential frequency optimisation'''
|-
| || '''Cis'''|| '''Trans'''
|-
| '''D-Space''' || {{DOI|10042/to-7523}}|| {{DOI|10042/to-7524}}
|-
| '''File type'''||.log||.log
|-
| '''Calculation Type'''||FREQ||FREQ
|-
| '''Calculation Method'''||RB3LYP||RB3LYP
|-
| '''Basis Set'''||LANL2DZ with added d orbitals for phosphorous||LANL2DZ with added d orbitals for phosphorous
|-
|'''E(RB+HF-LYP)'''|| -623.693 Hatree ± 3.808x10-3||-623.694 Hatree ± 3.808x10-3
|-
| '''RMS Gradient Norm'''|| 0.000 Hatree ± 3.808x10-3 || 0.000 Hatree ± 3.808x10-3
|-
| '''Imaginary Freq'''|| 0 || 0
|-
| '''Dipole moment'''|| 0.0731 debye ± 0.005|| 0.2293 ± 0.005
|-
| '''Point group'''||C2||CS
|-
| '''Job time'''|| 21 minutes 39.0 seconds|| 22 minutes 47.5 seconds
|} '''''Figure 24:''''' ''A table to show the summary of the calculation

No negative frequencies were seen, confirming the minima. Note: The vibrational modes themselves are discussed in a subsequent section.

In order to check that a reasonable structure had been reached, the bond lengths and angles for the trans isomer were measured and compared to Mo(CO)4(PPh3)2 literature values. This is shown in '''Figure 25'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Bond length'''|| '''Measured values for Mo(CO)4(PCl3)2 ± 0.005 /Å'''|| '''Literature values for Mo(CO)4(PPh3)2'''<ref name="Trans_MO_COMPLEX_PARAMETERS">G. Hogarth and T. Norman, ''Inorganic Chemistry Acta'' 1997, '''254''', 167 {{DOI|10.1016/S0020-1693(96)05133-X|}</ref>
|-
| Mo-P || 2.42|| 2.500
|-
| Mo-C || 2.06|| 2.005
|-
| C-O || 1.17|| 1.16
|-
| '''Bond angle '''|| '''Measured ± 0.05 /°'''|| '''Literature'''<ref name="Trans_MO_COMPLEX_PARAMETERS">G. Hogarth and T. Norman, ''Inorganic Chemistry Acta'' 1997, '''254''', 167 {{DOI|10.1016/S0020-1693(96)05133-X|}</ref>
|-
| C-Mo-C (trans) || 180 and 178.4 || 180.0
|-
| C-Mo-C (cis) || 89.2 and 90.8 || 92.1
|-
| P-Mo-C || 90.0, 88.4 and 91.6 || 87.2
|} '''''Figure 25:''''' ''A table to compare structural parameters to literature value parameters for a similar complex''

Comparison of the calculated parameters of Mo(CO)4(PCl3)2 with the literature values of Mo(CO)4(PPh3)2 shows the calculated parameters to be reasonable. The optimised complex is therefore a reasonable structure.

This calculation shows the trans isomer to be more stable than the cis isomer by 2.6 KJ mol-1. It is interesting to note that there is a disparity between this calculation and the previous calculation, which suggested that the cis isomer was more stable than the trans isomer by 2.6 KJ mol-1. For both calculations, the energy difference is still well within the order of error (10 KJ mol-1). This means no definitive conclusions can be drawn from these calculations as to which isomer is more stable. The small difference between the energies of the structural isomers suggests that there is a fine balance between cis and trans geometries of these complexes.

It is proposed that there are both steric and electronic contributions to stability. These two effects must be finely balanced between the two geometries for such a small energy difference to be seen.

'''Sterics:''' Significant steric clashes would be expected to exist in the cis-complex. These steric repulsions would be less severe in the trans complex and so the steric effect favours the trans isomer. It is important to note that this term could be reversed by introducing attractive interactions between ligands, which would favour the cis isomer.

'''Electronic:''' It is proposed that a structural trans effect<ref name="structuraltranseffect">B.J. Coe, S.J. Glenwright, ''Coordination Chemistry Reviews'' 2000, '''203''', 5-80{{DOI|10.1016/S0010-8545(99)00184-8}}</ref> exists in the cis-complex, which is not present in the trans-complex. This stabilises the cis-complex electronically. In the cis-complex, Mo-C bond lengths trans to phosphines are measured as 2.02Å ± 0.005. Mo-C bonds lengths cis to phosphines are measured as 2.05Å ± 0.005. In the trans complex, all M-C bond lengths are measured as 2.05Å ± 0.005. Whilst the trans effect is poorly understood, it can be inferred from bond lengths that there is some interaction between electron density on the phosphorous lone pair and the trans carbonyl ligands, which brings the carbonyl ligands in closer. An analogy can be drawn to synergistic bonding. It is important to note that it is highly unlikely that the phosphorous lone pair and the π* of a trans carbonyl ligand overlap directly due to the large distance between them, but some interaction must be present between these two orbitals, perhaps carried by metal based orbitals. This results in a stabilisation of the orbital bearing the lone pair and destabilisation of the unfilled carbonyl π*, resulting in electronic stabilisation of the complex as a whole. This is shown pictorially in '''Figure 26'''.

[[Image:STRUCTURALTRANSMO.png|thumb|centre|200px| '''Figure 26:''' ''Proposed orbital stabilisation by the structural trans effect. It is important to note that these orbitals do not overlap directly and the electron density from the Phosphorous must be carried some intermediate orbitals, most likely to be metal based orbitals.'']]

By varying the magnitudes of the steric and electronic components, one would expect to be able to tune the phosphine ligand to allow each structural isomer to be accessed. A rational design of the cis-complex is attempted in the following section.

====Rational design?====

Computational chemistry is often utilised in rational catalyst design. For instance, one structural isomer of a complex that is catalytically active may be more active than the other structural isomer. In order to maximise catalytic activity, the PR3 could be tuned in order to make one structural isomer dominate.

The trans isomer could be made to dominate easily by increasing the bulkiness of the R substituents, maximising steric repulsions in the cis isomer.

However, obtaining the cis isomer is more difficult and will be examined in more detail. The cis isomer could be achieved by increasing the electronic stabilisation of the cis isomer, whilst minimising sterics with a small, electron donating group. In order to further promote the cis isomer, attractive interactions between the two phosphine ligands could be implemented, for instance hydrogen bonding interactions or π stacking.

With this in mind, R = OH was modelled. It was imagined that hydrogen bonding interactions would exist between cis phosphines in the cis-complex, but would not be present in the trans-complex. Furthermore, structural trans-effects may exist with carbonyls trans to the electron donating phosphines. Finally, sterics were minimised with P(OH)3 being significantly smaller than PCl3 as shown in '''Figure 27'''.

[[Image:SpacefillingmodelsofClandOH.png|thumb|centre|400px| '''Figure 27:''' ''Simple space filling models of the PCl3 ligand and the P(OH)3 ligand'']]

The cis and trans Mo(CO)4(P(OH)3)2 complexes were optimised using DFT-B3LYP and a LANL2DZ basis set with added d orbitals for phosphorous. The output files were checked and a successful convergence was found for both isomers. The optimised structures are shown below. The summary of the optimisations is shown in '''Figure 28'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:CISCOMPLEXOHS.png|thumb|left|130px|'''The cis complex'''
<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_OHs.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>]]
|
|[[Image:TransComplexOHs.png|thumb|left|100px|'''The trans complex'''
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Trans_OH.mol</uploadedFileContents>
<text>Jmol</text>
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{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="3" |'''Rational Design of a stable cis conformer'''
|-
| || '''Cis'''|| '''Trans'''
|-
| '''D-Space''' || {{DOI|10042/to-7505}}|| {{DOI|10042/to-7506}}
|-
| '''File type'''||.log||.log
|-
| '''Calculation Type'''||FOPT||FOPT
|-
| '''Calculation Method'''||RB3LYP||RB3LYP
|-
| '''Basis Set'''||LANL2DZ with added d orbitals for phosphorous||LANL2DZ with added d orbitals for phosphorous
|-
|'''E(RB+HF-LYP)'''|| -989.063 Hatree ± 3.808x10-3||-989.048 Hatree ± 3.808x10-3
|-
| '''RMS Gradient Norm'''|| 0.000 Hatree ± 3.808x10-3 || 0.000 Hatree ± 3.808x10-3
|-
| '''Dipole moment'''|| 5.06 debye ± 0.005|| 1.71 debye ± 0.005
|-
| '''Point group'''||C1||C1
|-
| '''Job time'''|| 6 hours 12 minutes 10.8 seconds|| 8 hours 10 minutes 20.3 seconds
|} '''''Figure 28:''''' ''A table to show the summary of the optimisations''

In order to confirm the minima, frequency analysis was utilised. A summary of the frequency calculations is shown in '''Figure 29'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="3" |'''Rational Design of a stable cis conformer'''
|-
| || '''Cis'''|| '''Trans'''
|-
| '''D-Space''' || {{DOI|10042/to-7519}}|| {{DOI|10042/to-7522}}
|-
| '''File type'''||.log||.log
|-
| '''Calculation Type'''||FREQ||FREQ
|-
| '''Calculation Method'''||RB3LYP||RB3LYP
|-
| '''Basis Set'''||LANL2DZ with added d orbitals for phosphorous||LANL2DZ with added d orbitals for phosphorous
|-
|'''E(RB+HF-LYP)'''|| -989.063 Hatrees ± 3.808x10-3|| 989. 048 Hatree ± 3.808x10-3
|-
| '''RMS Gradient Norm'''|| 0.000 Hatree ± 3.808x10-3 || 0.00 Hatree ± 3.808x10-3
|-
| '''Imaginary Frequencies'''|| 0 || 0
|-
| '''Dipole moment'''|| 5.0582 debye ± 0.005|| 1.7142 debye ± 0.005
|-
| '''Point group'''||C1||C1
|-
| '''Job time'''|| 58 minutes 54.2 seconds|| 57 minutes 20.0 seconds
|} '''''Figure 29:''''' ''A table to show the summary of the frequency calculation''

It was found that all frequencies were positive and thus a minima has been successfully converged to. The measured bond lengths (Mo-P ≈ 2.5Å and Mo-C ≈ 2Å) are reasonable in comparison to the previously established bond lengths - so plausible structures have been reached.

It can be seen that the cis isomer is 39.4 KJ mol-1 more stable than the trans isomer, so the objective of this exercise is achieved and the cis isomer is favoured significantly.

At the start of the exercise, the complex was designed with 3 features in mind:

* Minimise steric effects
* Maximise electronic effects
* Introduce attractive interactions between ligands to favour the cis

In order to evaluate how successful this design procedure has been, it's important to evaluate the resultant structures to see how many of these effects are operating. The stabilisation of the cis isomer could merely be the result of reduced sterics! Firstly, the electronic stabilisation will be examined.

In the cis-complex, Mo-C bond trans to phosphines are measured to have a bond length of 2.01Å ± 0.005. Then Mo-C bonds cis to phosphines are measured to have a bond length of 2.04Å ± 0.005. In the trans complex, all M-C bonds were measured to be 2.04Å ± 0.005. The shortening of Mo-C bonds trans to phosphines in the cis-complex is more severe when R = OH compared to when R = PCl3 (2.01Å cf 2.02Å). This suggests some success in trying to maximise this electronic stabilisation of the cis-complex. However, this reduction in bond length is very small, which suggests some improvement could be made here.

Secondly, the presence of hydrogen bonds is examined. It was intended to introduce hydrogen bonds between cis ligands, to stabilise the cis isomer. Hydrogen bonding distances are often between 1.8 and 2.6 Å <ref name="Hbond">L. Benco, D. Tunega, J. Hafner and H. Lischka, ''Journal of Physical Chemistry B'' 2001, '''105''', 10812-10817{{DOI|10.1021/jp0124802}}</ref> In the cis complex, there are two H-O interactions at a distance of 1.8Å bridging the two cis ligands, these are "external" to each phosphine. There are then two further "internal" interactions at a distance of 2.5Å within each phosphine. In the trans complex, there are only two "internal" hydrogen bonding interactions.

====Frequency Analysis of Cis and Trans Mo(CO)4(PCl3)2====

The IR frequencies were calculated using DFT-B3LYP and a LANL2-DZ basis set, with included d orbitals for the phosphorous atom and ultrafine electronic convergence conditions. Cis {{DOI|10042/to-7523}} and Trans {{DOI|10042/to-7524}}

=====Low Frequency Vibrations=====

A number of vibrations were observed to have a low frequency, these are shown in '''Figure 30''' (cis) and '''Figure 31''' (trans).

'''Cis complex'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''||| '''Form of the vibration'''|| '''Frequency (error in frequency) /cm-1'''|| '''Intensity'''
|-
| 1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Cis_Vibrations_PCl3_OUTFILE.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #1</text>
</jmolAppletButton>
</jmol>
|| 12 (± 1) || 0
|-
| 2 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Cis_Vibrations_PCl3_OUTFILE.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #2</text>
</jmolAppletButton>
</jmol>
|| 20 (± 2) || 20
|-
| 3 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Cis_Vibrations_PCl3_OUTFILE.out</uploadedFileContents>
<script>frame 5; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #3</text>
</jmolAppletButton>
</jmol>
|| 46 (± 5) || 0
|-
|} '''''Figure 30:''''' ''A table to show the low frequencies of the cis-complex''

'''Trans complex'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''||| '''Form of the vibration'''|| '''Frequency (error in frequency) /cm-1'''|| '''Intensity'''
|-
| 1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Trans_vibrations_PCl3_outfile.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #1</text>
</jmolAppletButton>
</jmol>
|| 5 (± 0.5) || 0
|-
| 2 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Trans_vibrations_PCl3_outfile.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #2</text>
</jmolAppletButton>
</jmol>
|| 7 (± 0.7) || 0
|-
| 3 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Trans_vibrations_PCl3_outfile.out</uploadedFileContents>
<script>frame 5; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #3</text>
</jmolAppletButton>
</jmol>
|| 40 (± 4) || 0
|-
|} '''''Figure 31:''''' ''A table to show the low frequencies of the trans complex'

It is noticed that all of the low frequency vibrations involve rotations about the Mo-P bonds. At room temperature (298.15 K), there is 2476 J mol-1 of energy (RT) available to the system. This corresponds to a wavenumber of around 207 cm-1. Vibrations and rotations which experience vibrations and rotations much lower than this value will be occur spontaneously at room temperature. This implies that the phosphine groups are freely rotating about the Mo-P bonds at room temperature in reality.

This explains why the point group of the structures has been incorrectly assigned as C1. Gaussian is not taking free rotation about Mo-P bonds into account when it is calculating the point group of the molecule and thus the symmetry is being broken by the 'static' phosphines. This leads to the C2 axis not being found in the cis complex and the C4 rotation axis not being found in the trans complex and thus an incorrect point group assignment in each case.

=====Carbonyl IR Stretches=====

From the calculated vibrational frequencies, the frequencies corresponding to carbonyl vibrations were found. The symmetry of the vibrations was assigned using a representation table as illustrated in '''Figure 9''' previously.

'''Cis complex'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''||| '''Form of the vibration'''|| '''Frequency (error in frequency) /cm-1'''|| '''Literature'''|| '''Intensity''' || '''Symmetry'''
|-
| 42 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Cis_Vibrations_PCl3_OUTFILE.out</uploadedFileContents>
<script>frame 44; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #42</text>
</jmolAppletButton>
</jmol>
|| 1939 (± 194) || 1986|| 1599 || B2
|-
| 43 ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Cis_Vibrations_PCl3_OUTFILE.out</uploadedFileContents>
<script>frame 45; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #43</text>
</jmolAppletButton>
</jmol>
|| 1942 (± 194) || 1994 || 820|| B1
|-
| 44 ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Cis_Vibrations_PCl3_OUTFILE.out</uploadedFileContents>
<script>frame 46; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #44</text>
</jmolAppletButton>
</jmol>
|| 1953 (± 195) || 2004|| 591|| A1
|-
| 45 ||<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Cis_Vibrations_PCl3_OUTFILE.out</uploadedFileContents>
<script>frame 47; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #45</text>
</jmolAppletButton>
</jmol>
|| 2019 (± 20) || 2072 || 542|| A1
|} '''''Figure 32:''''' ''A table to show the CO stretches for the cis complex''

'''Trans complex'''
{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''||| '''Form of the vibration'''|| '''Frequency (error in frequency) /cm-1'''|| '''Literature'''|| '''Intensity''' || '''Symmetry'''
|-
| 42 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Trans_Isomer_FREQ_CALC.out</uploadedFileContents>
<script>frame 44; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #42</text>
</jmolAppletButton>
</jmol>
|| 1939 (± 194) || 1896|| 1605 || EU
|-
| 43 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Trans_Isomer_FREQ_CALC.out</uploadedFileContents>
<script>frame 45; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #43</text>
</jmolAppletButton>
</jmol>
|| 1940 (± 194) || 1896|| 1606 || EU
|-
| 44 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Trans_Isomer_FREQ_CALC.out</uploadedFileContents>
<script>frame 46; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #44</text>
</jmolAppletButton>
</jmol>
|| 1967 (± 197) || - || 6 || B1g
|-
| 45 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Trans_Isomer_FREQ_CALC.out</uploadedFileContents>
<script>frame 47; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Stretch #45</text>
</jmolAppletButton>
</jmol>
|| 2026 (± 203) || - || 5 || A1g
|} '''''Figure 33:''''' ''A table to show the CO stretches for the cis complex''

The point group of the cis complex is C2V. If stretching vectors are assigned along each CO bond, one can determine the number of expected normal stretching modes. A representation table is set up for the stretching vectors and is shown in '''Figure 34'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''C2v|| '''E'''|| '''C2'''|| '''σv (xy)''' || '''σv(xz)'''
|-
| '''Γ''' [[Image:Bc608_Mo_CO_PCL3_Stretches.png|centre|100px]] || 4 || 0 || 2 || 2
|} '''''Figure 34''''' : ''The representation table for the cis complex''

Using the reduction formula it is found that 2 stretches of A1 symmetry, one stretch of B1 symmetry and one stretch of B2 symmetry are expected. This is indeed what is observed computationally.

The same analysis is then applied to the trans complex. The point group of the trans complex is D4h. Again, stretching vectors are assigned along each CO bond. A representation table is set up for the stretching vectors '''Figure 35'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''D4h|| '''E'''|| '''2C4'''|| '''C2''' || '''2C2' '''|| '''2C2<nowiki>''</nowiki>''' || '''i'''|| '''2S4'''|| '''σh''' ||'''2σv''' || '''2σd'''
|-
| '''Γ''' [[Image:Bc608_Mo_CO_PCL3_TRANS_STRETCHES.png|centre|100px]] || 4 || 0 || 0 || 2 || 0 || 0 || 0 || 4 || 2 || 0
|} '''''Figure 35''''' : ''The representation table for the trans-complex''

Using the reduction formula it is found that 1 stretch of A1g, 1 stretch of B1g and 1 doubly degenerate stretch of EU symmetry are expected, again predicting 4 vibrations. However, the B1g and A1g stretches are totally symmetric, thus no change in dipole moment occurs throughout the vibration and so the totally symmetric vibrations will not be IR active. One would therefore expect 1 peak to be seen in the IR spectrum of EU symmetry corresponding to 2 degenerate vibrations.

4 calculated vibrational modes are seen. However the A1g and B1g vibrational modes are of very low intensity. The expected EU vibrations are degenerate and thus would give one peak in an IR spectrum. The reason for the A1g and B1g vibrations having a non-zero intensity is because the molecule has deviations from ideal bond angles. The C-Mo-C bond angles were not measured to be 90° as one would expect them to be, this leads to a small computed change in dipole moment for the A1g and B1g vibrations.

===Mini project: Bent's rules applied to simple carbene systems===

====Introduction====

In 1960, Bent proposed a rule in order to justify deviations in structures from VSEPR theory.<ref name="Bent">H.A. Bent, ''Chemical Reviews'' 1961, '''61''', 275-311 {{DOI|10.1021/cr60211a005}}</ref> Bent's rule is a hybridisation argument that states:

''"Atomic s character tends to concentrate in orbitals that are directed toward electropositive groups and atomic p character tends to concentrate in orbitals that are directed toward electronegative groups”'' <ref name="Bent">H.A. Bent, ''Chemical Reviews'' 1961, '''61''', 275-311 {{DOI|10.1021/cr60211a005}}</ref>

Bent's rules have been applied to carbenes ('''Figure 36''') and carbene like species in order to predict substituent effects on spin multiplicity. For a carbene substituted by electronegative substituents, the p character on the atomic carbon orbitals is directed towards the substituents, imparting a greater s character on the carbene electrons, leading to singlet carbenes. For a carbene substituted by electropositive substituents, the s character on the atomic carbon orbitals is directed towards the substituents, imparting a greater p character on the carbene electrons, leading to triplet carbenes.

[[Image:Bc608_Cartoonofcarbenes.png|thumb|centre|300px|'''''Figure 36:''''' ''A cartoon to show a) A bent singlet carbene, b) A bent triplet carbene and c) an idealised linear triplet carbene]]

Bent's rules are based on empirical observations and have little theoretical basis. The aims of this project are as follows:

* To recreate empirical observations ''in silico''
* To use NBO analysis to examine if the hybridisation of the carbenes does change with electronegativity of substituents.
* To calculate the molecular orbitals of the system, to see if there is a Molecular Orbital basis for the observed trends.
* If possible, to provide a rationalisation of Bent's rules when applied to predictions of carbene spin multiplicity.

====Optimisation of Structures====

A series of structures of the type CR2 were optimised using DFT-B3LYP restricted-open shell calculations. The calculation was restricted to pair up α and β wavefunctions, as individual α and β wavefunctions are unfamiliar and would be difficult to interpret.

A 6-31G basis set is used to model most compounds. Less accurate basis sets have been previously used to calculate more complex systems, thus based on the limited experience gained throughout this lab, it is believed that the 6-31G basis set will suffice for the modelling of simple 3 atom carbene systems. Compounds containing second row elements are modelled with a 6-311G(d,p) basis set to account for the polarizability and diffusivity of orbitals on these second row elements. '''Figure 37''' shows the optimised structures of the compounds.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''R'''|| '''Basis Set'''|| '''D-Space/Log file Singlet''' || '''D-Space/Log file Triplet''' || '''Energy of Singlet ± 3.808x10-3 / Hatree''' || '''Energy of Triplet ± 3.808x10-3 / Hatree ''' || '''Structure Singlet''' || '''Structure Triplet'''
|-
| H ||6-31G || https://wiki.ch.ic.ac.uk/wiki/images/9/93/Bc608_CH2_SINGLET_GEO_OPT.LOG|| https://wiki.ch.ic.ac.uk/wiki/images/3/32/Bc608_CH2_TRIPLET_GEO_OPT.LOG || -39.114 || -39.142||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CH2_SINGLET.mol</uploadedFileContents>
<text>CH2 Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CH2_TRIPLET.mol</uploadedFileContents>
<text>CH2 Triplet</text>
</jmolAppletButton>
</jmol>
|-
| SiH3 ||6-311G(d,p) || {{DOI|10042/to-7622}}|| {{DOI|10042/to-7623}} || -620.605 || -620.641 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SIH3_SINGLET.mol</uploadedFileContents>
<text>C(SiH3)2 Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SIH3_TRIPLET.mol</uploadedFileContents>
<text>C(SiH3)2 Triplet</text>
</jmolAppletButton>
</jmol>
|-
| NH3+ || 6-31G || {{DOI|10042/to-7620}} || {{DOI|10042/to-7621}} || -150.287 || -150.296 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NH3_SINGLET.mol</uploadedFileContents>
<text>C(NH3)22+ Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NH3_TRIPLET.mol</uploadedFileContents>
<text>C(NH3)22+ Triplet</text>
</jmolAppletButton>
</jmol>
|-
| NH2 || 6-31G || https://wiki.ch.ic.ac.uk/wiki/images/4/41/BC608_NH2_GEOOPT_SINGLET.LOG || https://wiki.ch.ic.ac.uk/wiki/images/f/f7/Bc608_NH2_GEOOPT_TRIPLET.LOG || -149.919 || -149.798 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NH2_SINGLET.mol</uploadedFileContents>
<text>C(NH2)2 Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NH2_TRIPLET.mol</uploadedFileContents>
<text>C(NH2)2 Triplet</text>
</jmolAppletButton>
</jmol>
|-
| F || 6-31G || https://wiki.ch.ic.ac.uk/wiki/images/c/cf/Bc608_CF2_SINGLET_GEO_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7d/Bc608_CF2_TRIPLET_GEOOPT.LOG || -237.625 || -237.552 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CF2_SINGLET.mol</uploadedFileContents>
<text>CF2 Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CF2_TRIPLET.mol</uploadedFileContents>
<text>CF2 Triplet</text>
</jmolAppletButton>
</jmol>
|-
| Br || 6-31G ||https://wiki.ch.ic.ac.uk/wiki/images/1/14/Bc608_BR2_SINGLET_GEO_OPT.LOG || https://wiki.ch.ic.ac.uk/wiki/images/1/12/BC608_BR2_TRIPLET_GEO_OPT.LOG|| -5181.112 || -5181.088 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BR2_SINGLET.mol</uploadedFileContents>
<text>CF2 Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BR2_TRIPLET.mol</uploadedFileContents>
<text>CF2 Triplet</text>
</jmolAppletButton>
</jmol>
|}'''Figure 37''': ''The optimised structures of the type CR2 using DFT-B3LYP and the indicated basis sets''

All optimisations were found to converge to a stationary point.

====Frequency Analysis to confirm a minima====

In order to confirm the minima, the frequencies were calculated using the same basis sets as the optimisations. The results are shown in '''Figure 38'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''R'''|| '''Basis Set'''|| '''D-Space/Log file Singlet''' ||'''D-Space/Log file Triplet''' || '''Low Frequencies singlet /cm-1''' || '''Lowest 'real' frequency singlet /cm-1''' || '''Low Frequencies triplet /cm-1''' || '''Lowest 'real' frequency triplet /cm-1''' ||
|-
| H || 6-31G|| {{DOI|10042/to-7624}} || {{DOI|10042/to-7625}} || -45, -43, -18, 0, 0, 0 || 1428 || 0, 0, 0, 16, 17, 33 || 1124
|-
| SiH3 || 6-311G(d,p)|| {{DOI|10042/to-7626}} || {{DOI|10042/to-7627}} || -19, -9, -8, 0, 0, 2 || 114 || -6, -6, -5, 0, 0, 0 || 32
|-
| NH3+ || 6-31G || {{DOI|10042/to-7628}} || {{DOI|10042/to-7629}} || -22, -11, -11, 0, 0, 0 || 134 || -19 -6, 0, 0, 0, 3 || 114
|-
| NH2 || 6-31G || {{DOI|10042/to-7630}} || {{DOI|10042/to-7632}} || -14, 0, 0, 0 12, 24 || 520 || -768, -620, -474, -385, -13, 0 || 6.6
|-
| F || 6-31G || {{DOI|10042/to-7618}} || {{DOI|10042/to-7619}} || -27, -17, -15, 0, 0, 0 || 581 || - 10, 0, 0, 0 12, 13 || 464
|-
| Br || 6-31G || {{DOI|10042/to-7634}} || {{DOI|10042/to-7635}} || -40, -15, -11, 0, 0, 0 || 174 || -43, -11, 0, 0, 0, 6 || 166
|}'''Figure 38:''' ''The frequency analyses of the previously optimised structures

The frequency analysis was somewhat disappointing! The R = NH2 triplet showed worryingly negative low frequencies and a number of structures showed low frequencies outside the ideal 10 to -10 cm-1 range. Due to the approaching deadline and a big queue on SCAN, the structures to be modelled need to be reconsidered so that the project can be realistically achieved in the timeframe available. The effects of substituent electropositivity can be examined by going down group 17. The elements of group 17 also provide an added advantage of being typically monovalent in their bonding, thus avoiding complex structures. Furthermore, the low frequencies for these calculations were reasonable previously with a 6-31G basis set and should improve with a higher basis set. Most importantly, they can be modelled on desktop computers thus avoiding the SCAN queue.

====Reoptimisation and frequency analysis using an improved basis set====

A series of structures of the type CR2 were optimised using DFT-B3LYP restricted-open shell calculations with a 6-311++G(d,p) basis set. '''Figure 39''' shows the optimised structures of the compounds.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''R'''||'''Log file Singlet''' || '''Log file Triplet''' || '''Energy of Singlet ± 3.808x10-3 / Hatree''' || '''Energy of Triplet ± 3.808x10-3 / Hatree ''' || '''Structure Singlet''' || '''Structure Triplet'''
|-
| H ||https://wiki.ch.ic.ac.uk/wiki/images/7/77/Bc608_improvedbasis_H_SINGLET_GEOOPT.LOG || https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_improvedbasis_H_TRIPLET_GEOOPT.LOG ||-39.147|| -39.164||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_impbas_H_SINGLET.mol</uploadedFileContents>
<text>Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_impbas_H_TRIPLET.mol</uploadedFileContents>
<text>Triplet</text>
</jmolAppletButton>
</jmol>
|-
| F ||https://wiki.ch.ic.ac.uk/wiki/images/4/4f/Bc608_improvedbasis_F_SINGLET_GEOOPT.LOG|| https://wiki.ch.ic.ac.uk/wiki/images/5/57/Bc608_improvedbasis_F_TRIPLET_GEOOPT.LOG|| -237.774 ||-237.690||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_impbas_F_SINGLET_GEOOPT.mol</uploadedFileContents>
<text>Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_impbas_F_TRIPLET_GEOOPT.mol</uploadedFileContents>
<text>Triplet</text>
</jmolAppletButton>
</jmol>
|-
| Cl || https://wiki.ch.ic.ac.uk/wiki/images/d/d1/Bc608_improvedbasis_CL_SINGLET_GEOOPT.LOG|| https://wiki.ch.ic.ac.uk/wiki/images/3/3f/Bc608_improvedbasis_CL_TRIPLET_GEOOPT.LOG|| -958.454 || -958.423 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CL_SINGLET_GEOOPT.mol</uploadedFileContents>
<text>Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CL_TRIPLET_GEOOPT.mol</uploadedFileContents>
<text>Triplet</text>
</jmolAppletButton>
</jmol>
|-
| Br || https://wiki.ch.ic.ac.uk/wiki/images/a/a0/Bc608_improvedbasis_BR_SINGLET_GEOOPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/5/53/Bc608_improvedbasis_BR_TRIPLET_GEOOPT.LOG|| -5186.295 || -5186.270 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BR_SINGLET_GEOOPT.mol</uploadedFileContents>
<text>Singlet</text>
</jmolAppletButton>
</jmol>
||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BR_TRIPLET_GEOOPT.mol</uploadedFileContents>
<text>Triplet</text>
</jmolAppletButton>
</jmol>
|}'''Figure 39:''' ''Optimised structures of the type CR2 using DFT-B3lYP and a 6-311++G(d,p) basis set.

A frequency analysis was then performed on the optimised structures in order to confirm minima.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''R'''|| '''D-Space/Log file Singlet''' ||'''D-Space/Log file Triplet''' || '''Low Frequencies singlet /cm-1''' || '''Lowest 'real' frequency singlet /cm-1''' || '''Low Frequencies triplet /cm-1''' || '''Lowest 'real' frequency triplet /cm-1''' ||
|-
| H || https://wiki.ch.ic.ac.uk/wiki/images/4/4c/Bc608_H_SINGLET_FREQ.LOG|| https://wiki.ch.ic.ac.uk/wiki/images/b/b0/Bc608_H_TRIPLET_FREQ.LOG|| -46, -38, -36, 0, 0, 0 || 1385 || 0, 0, 0, 47, 52, 53 || 1083
|-
| F || https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_F_SINGLET_FREQ.LOG||https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_F_TRIPLET_FREQ.LOG|| -36, -29, -21, 0, 0, 0 || 668 || -27, -22, -11, 0, 0, 0 || 509
|-
| Cl || https://wiki.ch.ic.ac.uk/wiki/images/4/43/Bc608_CL_SINGLET_FREQ.LOG|| https://wiki.ch.ic.ac.uk/wiki/images/6/62/Bc608_CL_TRIPLET_FREQ.LOG|| 0, 0, 0, 12, 12, 34 || 329 || - 15, -6, 0, 0, 0, 9 || 300
|-
| Br || https://wiki.ch.ic.ac.uk/wiki/images/c/c8/Bc608_BR_SINGLET_FREQ.LOG|| https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BR_TRIPLET_FREQ.LOG|| 0, 0, 0, 3, 6, 7 || 190 || 0, 0, 0, 4, 7, 27 || 183
|}'''Figure 40:''' ''Frequency Analyses of the previously optimised structures.''

As can be seen in '''Figure 40''', the low frequencies were much more acceptable than the last set of calculations. The low frequencies were all at least an order of magnitude smaller than the lowest 'real' frequency in all cases. No negative frequencies were seen for the normal stretching modes confirming minima. The results of vibrational analyses are compared to literature in a later section. However, the analysis shows excellent agreement between calculated stretching frequencies and literature, so plausible optimised structures have been reached.

It can be seen that the triplet is favoured for the electropositive H substituent, whereas the singlet is favoured for the electronegative halogen substituents - as predicted by Bent's rules. Now the empirical observations have been recreated, an attempt will be made to understand why electronegativity affects spin multiplicity. Bent's rules are an argument based upon bond hybridisation, therefore an NBO analysis of the structures seems a logical place to start.

====NBO Analysis====

Core to Bent's rules is a hybridisation argument which is substituent electronegativity dependent. In order to test whether there is any basis to this argument, it would be desirable to calculate the hybridisation of the Natural Bond Orbitals in which the carbene electrons reside. The Natural Bond Orbitals of the singlets of CH2, CF2, CCl2 and CBr2 had been calculated in the previous frequency analyses.

Interestingly, NBO calculations imply that every singlet carbene modelled is ionic in character. No 2 centre - 2 electron bonding orbitals were found in the NBO analysis, denoted in the log file as "BD". Instead, all occupied orbitals are either core ("CR") or lone pair ("LP") orbitals, with higher orbitals being diffuse Rydberg ("Ry") orbitals.

Conversely, the triplet states of these carbenes showed bonding orbitals in all cases, with the Singularly Occupied Molecular Orbitals being a carbon orbital of 100% p character and another carbon based orbital of varying character referred to as "carbene orbital" here-on-in. '''Figure 40''' shows higher NBO charges for the singlet when compared to the triplet, supporting the ionic bonding proposed in the singlet species.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''R'''||'''NBO charge on R in Singlet''' ||'''NBO charge on C in Singlet'''|| '''NBO charge on R in Triplet'''||'''NBO charge on C in Triplet'''
|-
| H || 0.527 || 2.946 || 0.113 || -0.226
|-
| F || 4.335 || 3.331 || -0.345 || 0.690
|-
| Cl || 8.530 || 2.940 || 0.070 || -0.140
|-
| Br || 17.569 || 2.862 || 0.162 || -0.325
|}'''Figure 40:''' ''A comparison of NBO charges of CR2 fragments.''

[[Image:Bc608_F_SINGLET_NBOCHARGE.png|thumb|centre|300px|'''''Figure 41:''''' ''A diagram to show the NBO charges on singlet CH2. It can be seen that the carbon is highly electron deficient.]]

Whilst it is very interesting that the two different state multiplicities exhibit completely different bonding types, further calculations in which the Lewis structure is fixed would be required to try and understand why this is the case. Interpretation of the singlet NBO log files proved far too complex for a beginner to understand, with the expected "carbene orbital" not being found. Instead, 4 NBOs of varying character were seen. An example of these orbitals is shown in '''Figure 42'''.

12. (0.99553) LP*( 1) C 1 s( 73.70%)p 0.36( 26.21%)d 0.00( 0.09%)
0.0000 0.8580 0.0293 0.0002 0.0001
0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.5102 0.0407
0.0110 -0.0022 0.0000 0.0000 0.0000
-0.0061 -0.0294
13. (0.44348) LP*( 2) C 1 s( 27.31%)p 2.65( 72.32%)d 0.01( 0.37%)
0.0000 0.5107 -0.1108 -0.0009 -0.0005
0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 -0.8504 -0.0016
0.0109 0.0021 0.0000 0.0000 0.0000
-0.0413 0.0448
14. (0.37224) LP*( 3) C 1 s( 0.00%)p 1.00( 99.50%)d 0.01( 0.50%)
0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.9939
-0.0848 -0.0001 0.0008 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 -0.0706
0.0000 0.0000
15. (0.23875) LP*( 4) C 1 s( 0.00%)p 1.00( 99.52%)d 0.00( 0.48%)
0.0000 0.0000 0.0000 0.0000 0.0000
0.9968 0.0403 -0.0058 0.0068 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 -0.0690 0.0000
'''Figure 42:''' ''Partially filled NBO orbitals where the carbene lone pair should occur.''

Whilst the singlet NBO analysis proved complex, the hybridisation of triplet orbitals was much more straightforward. The hybridisation of triplet orbitals is examined in '''Figure 43'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''R'''||'''S character / %''' || '''P character /%'''
|-
| H || 33.98% || 66.02%
|-
| F || 63.59% || 36.39%
|-
| Cl || 59.17% || 40.81%
|-
| Br || 63.43% || 36.55%
|}'''Figure 43:''' ''A comparison of hybridisation for the NBOs of triplet CR2 fragments.''

From '''Figure 43''' it would appear that Bent's hybridisation arguments are applicable. CH2 shows sp2 hybridisation, of the "carbene orbital", whereas more electronegative groups show increased s character. When this "carbene" orbital has high p character, it will be close in energy to the non-bonding carbon p orbital and thus the triplet state will be favoured. A high s character will stabilise the "carbene orbital" by bringing the electrons closer to the nucleus. This would increase the energy gap between the non bonding p orbital and the "carbene orbital". The spin pairing energy will then be smaller than the promotion energy and the singlet state will be favoured. This is shown in '''Figure 44'''.

[[Image:Bc608_cARBENEORBITALDIAGRAMSPINPAIRVSPROMOTE.png|thumb|centre|600px|'''''Figure 44:''''' ''A diagram to show spin pairing against promotion energy for high s character "carbene orbitals" and high p character "carbene orbitals" ''.]]

====IR Stretching frequencies of dihalocarbenes: A comparison of bond strengths====

The lack of bonding interactions in the Natural Bond Orbital calculations of CR2 singlets was unexpected and the possibility of the modelled structures poorly reflecting reality was entertained. In order to see if the calculated bonds are reasonable, the vibrational spectra of the dihalo compounds are examined in detail. Unfortunately, due to the instability of the CH2 diradical, no literature vibrational frequencies or bond lengths could be found and the plausibility of the optimised structure could not be determined.

=====CCl2=====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''|| '''Form of the vibration'''|| '''Frequency (error in frequency) /cm-1'''|| '''Intensity''' || '''Symmetry within the C2V point group'''
|-
| 1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CL_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Symmetric Scissoring</text>
</jmolAppletButton>
</jmol>
|| 329 (± 33) || 1|| A1
|-
| 2 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CL_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 4; vectors 4; rotate 90; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Symmetric Stretching</text>
</jmolAppletButton>
</jmol>
|| 709 (± 71) || 42 || A1
|-
|3 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_CL_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 5; vectors 4; vectors scale 1.5; rotate 90; color vectors white; vibration 4;</script>
<text>Asymmetric Stretching</text>
</jmolAppletButton>
</jmol>
|| 711 (± 71)|| 429 || B1
|} '''''Figure 44:''''' ''A table to show the IR stretches of CCl2''

Experimentally, 2 IR absorptions are observed at 748 and 721 cm-1<ref name="CCL2IRFREQ">D.E. Milligan and M.E. Jacox, ''The Journal of Chemical Physics'' 1967, '''47''', 703-707 {{DOI|10.1063/1.1711942}}</ref>. The symmetric scissoring A1 stretch shows no change in dipole moment and thus is not observed experimentally. There is a very strong agreement between the calculated IR values and the literature IR values, suggesting that the optimised structure of CCl2 is reasonable.

=====CBr2=====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''|| '''Form of the vibration'''|| '''Frequency (error in frequency) /cm-1'''|| '''Intensity''' || '''Symmetry within the C2V point group'''
|-
| 1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BR_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Symmetric Scissoring</text>
</jmolAppletButton>
</jmol>
|| 190 (± 19) || 1|| A1
|-
| 2 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BR_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 4; vectors 4; rotate 90; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Symmetric Stretching</text>
</jmolAppletButton>
</jmol>
|| 594 (± 59) || 18 || A1
|-
|3 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BR_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 5; vectors 4; vectors scale 1.5; rotate 90; color vectors white; vibration 4;</script>
<text>Asymmetric Stretching</text>
</jmolAppletButton>
</jmol>
|| 624 (± 62)|| 368 || B1
|} '''''Figure 45:''''' ''A table to show the IR stretches of CBr2''

Experimentally, 2 IR absorptions are observed at 595 and 640 cm-1<ref name="CBr2IRFREQ">L. Andrews and T. Granville Carver, ''The Journal of Chemical Physics'' 1968, '''49''', 896-902 {{DOI|10.1063/1.1670158}}</ref>. Again, the symmetric scissoring A1 stretch shows no change in dipole moment and thus is not observed experimentally. Again, there is a very strong agreement between the calculated IR values and the literature IR values, suggesting the optimisation of CBr2 has given a reasonable structure.

=====CF2=====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Stretching Mode #'''|| '''Form of the vibration'''|| '''Frequency (error in frequency) /cm-1'''|| '''Intensity''' || '''Symmetry within the C2V point group'''
|-
| 1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_F_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Symmetric Scissoring</text>
</jmolAppletButton>
</jmol>
|| 668 (± 67) || 4|| A1
|-
| 2 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_F_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 4; vectors 4; rotate 90; vectors scale 1.5; color vectors white; vibration 4;</script>
<text>Asymmetric Stretching</text>
</jmolAppletButton>
</jmol>
|| 1086 (± 109) || 423 || B1
|-
|3 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_F_SINGLET_FREQ.LOG</uploadedFileContents>
<script>frame 5; vectors 4; vectors scale 1.5; rotate 90; color vectors white; vibration 4;</script>
<text>Symmetric Stretching</text>
</jmolAppletButton>
</jmol>
|| 1209 (± 121)|| 136 || A1
|} '''''Figure 46:''''' ''A table to show the calculated vibrational frequencies of CF2''

No reports of the isolation or characterisation of diflurocarbene could be found, thus a direct comparison to literature bond lengths, angles or IR spectra is not possible. However, the stretching frequencies can be inferred from literature data for CCl2. C-F bonds will be stronger than C-Cl bonds due to increased overlap between fluorine valence orbitals and carbon valence orbitals, than between chlorine valence orbitals and carbon valence orbitals. One would therefore expect higher vibrational frequencies for difluorocarbene compared to the dichlorocarbene. Stretching frequencies in the 1000-1200 cm-1 range therefore seem reasonable, implying a reasonable structure of the CF2 carbene has been reached.

====Molecular Orbital Examination of the Dihalocarbenes====

=====Effects of electronegativity=====

The NBO analysis was not entirely successful in explaining Bent's rules and the analysis limited by the unexpected and interpretable singlet species results. The molecular orbitals were examined in order to see if they could provide any rationalisation of Bent's rules. '''Figure 47''' shows the qualitative Molecular Orbitals of CF2 as determined by LCAO.

[[Image:Bc608_CF2_FULL_MO_DIAGRAM.png|thumb|centre|1000px| '''Figure 47:''' ''The qualitative MO diagram for CF2'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Symmetry Label'''|| '''Qualitative MO'''||'''Calculated MO'''||'''Singlet Energy /Hatree'''||'''Triplet Energy /Hatree'''
|-
| 3a1 || [[Image:Bc608_3a1.png|centre|100px|]]|| [[Image:Bc608_MO4.png|centre|100px|]] ||-1.32204 || -1.28124
|-
| 2b1 || [[Image:Bc608_2b1.png|centre|100px|]]|| [[Image:Bc608_MO5.png|centre|100px|]] || -1.25135 || -1.24266
|-
| 4a1 || [[Image:Bc608_4a1.png|centre|100px|]]|| [[Image:Bc608_MO6.png|centre|100px|]] || -0.69741 || -0.66100
|-
| 3b1 || [[Image:Bc608_3b1.png|centre|100px|]]|| [[Image:Bc608_MO7.png|centre|100px|]] || -0.57790 || -0.56602
|-
| 1b2 || [[Image:Bc608_1b2.png|centre|100px|]]|| [[Image:Bc608_Mo8.png|centre|100px|]] || -0.53523 || - 0.50434
|-
| 5a1 || [[Image:Bc608_5a1.png|centre|100px|]]|| [[Image:Bc608_MO9.png|centre|100px|]] || -0.52652 || - 0.50350
|-
| 1a2 || [[Image:Bc608_1a2.png|centre|100px|]]|| [[Image:Bc608_MO10.png|centre|100px|]] || -0.46641 || -0.45140
|-
| 4b1 || [[Image:Bc608_4b1.png|centre|100px|]]|| [[Image:Bc608_MO11.png|centre|100px|]] || -0.44492 || -0.44690
|-
| 6a1 ||[[Image:Bc608_6a1.png|centre|100px|]]|| [[Image:Bc608_MO12.png|centre|100px|]] || -0.32227 || -0.17831
|-
| 2b2 ||[[Image:Bc608_2b2.png|centre|100px|]]|| [[Image:Bc608_MO13.png|centre|100px|]] || -0.11837 || - 0.08822
|}
'''Figure 48:''' ''A table to compare calculated quantitative MOs for singlet CF2 with qualitative MOs''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''R'''|| '''HOMO-2''' ||'''HOMO-1'''||'''HOMO'''||'''LUMO'''||'''HOMO-LUMO Gap / KJ mol-1'''
|-
| F || [[Image:Bc608_MO10.png|centre|100px|]] ||[[Image:Bc608_MO11.png|centre|100px|]]|| [[Image:Bc608_MO12.png|centre|100px|]]||[[Image:Bc608_MO13.png|centre|100px|]] || 535
|-
| Cl || [[Image:Bc608_Cl_HOMO_2.png|centre|100px|]] || [[Image:Bc608_HOMO_1_CL.png|centre|100px|]] || [[Image:Bc608_Cl_HOMO.png|centre|100px|]] || [[Image:Bc608_Cl_LUMO.png|centre|100px|]]|| 365
|-
| Br || [[Image:Bc608_HOMO_2_BR.png|centre|100px|]]||[[Image:Bc608_HOMO_1_BR.png|centre|100px|]] ||[[Image:Bc608_HOMO_BR.png|centre|100px|]] ||[[Image:Bc608_Br_LUMO.png|centre|100px|]] || 326
|} '''Figure 49:''' ''A table to compare the orbitals between the HOMO -2 and the LUMO for CF2, CCl2 and CBr2''

'''Figure 49''' shows a number of interesting points. Firstly, the HOMO-LUMO gap is decreasing as group 17 is descended. This is decreasing the promotion energy required to generate a triplet state. As the group is descended the elements become more electropositive and more sterically bulky. This is then in excellent agreement with Bent's rules, which state electropositive and bulky substituents give triplet carbenes.

'''Figure 49''' also provides some indication as to why electronegativity of substituents effects spin multiplicity.

When two fragment orbitals (FOs) combine, they produce two molecular orbitals (MOs) - an in-phase, bonding combination of FOs and an out-of-phase antibonding combination of FOs. The relative contribution of each FO to each MO is determined by the FOs proximity in energy to the MO i.e. the closer in energy an FO is to an MO, the higher the contribution of that FO to that MO. Electronegative FOs are lower in energy than electropositive FOs. This means that a bonding MO will have an increased contribution from the more electronegative FO, whereas the corresponding antibonding MO will have an increased contribution from the more electropositive FO.

In the case of dichlorocarbenes, the HOMO is comprised of an antibonding combination of the carbon 2s orbital and a dihalogen FO derived from pZ orbitals (see '''Figure 48'''). The more electropositive the halogen, the higher in energy this dihalogen FO will be. It will therefore be closer in energy to the HOMO relative to an electronegative halogen. This leads to increased contribution from the dihalogen FO to the HOMO. This results in increased p character and diminished s character of the HOMO or "carbene orbital". The LUMO in all cases is derived from a dihalogen FO that was originally degenerate with the dihalogen FO that contributes to the HOMO.

The more electropositive the element is therefore, the more the HOMO and LUMO will resemble the unmixed dihalogen FOs and therefore the closer they will be in energy. They will be closer in energy because the dihalogen FOs being degenerate in the unmixed state, thus the HOMO-LUMO gap decreases.

=====Effects of sterics=====

It is proposed that bulky substituents will make the molecule more linear due to steric repulsion between substituents. Unfortunately, Bromine was not bulky enough to cause sufficient linearisation of the molecule for the energy ordering of the molecular orbitals to change significantly enough for the triplet to be favoured. However, a Walsh correlation diagram can be used to imagine what would happen to the ordering of the molecular orbitals upon the molecule becoming more linear. This is shown in '''Figure 50'''.

[[Image:Bc608_WalshCorrelationDiagramForCR2.png|thumb|centre|600px|'''''Figure 50:''''' ''A Walsh correlation diagram to illustrate how the ordering of molecular orbital energies would change with linearisation. Symmetry labels are retained throughout the transformation in order to make it easier to see where each molecular orbital has gone.'']]

It can be seen that the formerly very antibonding 7a1 molecular orbital has become relatively non-bonding upon linearisation, making this orbital become the new HOMO. The LUMO, 2b2 remains the LUMO, but is increased in energy due to diminished constructive overlap between the two pZ orbitals in the R2 fragment. The former HOMO, 6a1 orbital becomes less antibonding upon linearisation and is lowered in energy to become the HOMO-1. Antibonding interactions in the 4b1 are also reduced, by increasing the distance between pX orbitals in the R2 fragment. The HOMO and LUMO will be very close in energy and thus promotion will be preferred over spin pairing, thus the triplet is favoured for structures showing more linear behaviour.

Further calculations using bulky substituents such as tertiary butyl or SiMe3 would be required to confirm this effect. Unfortunately, there was insufficient time to perform these calculations within this report.

====Conclusion of the mini project====

In conclusion, the empirically observed spin multiplicity of substituted carbenes has been recreated ''in silico''. Interestingly, NBO analysis indicates that singlet carbenes are mostly ionic in their bonding, whereas triplet species have more covalent character. The cause of these observed differences however remains elusive. Further work in this area is required.

The main initial aim of this work was to propose a rationalisation of Bent's rules applied to simple carbene systems and this has been achieved.

'''''Electronegative''' substituents have a small contribution to the HOMO, leading to a high s character at the carbon centre and a large HOMO-LUMO gap, which favours spin pairing over electron promotion, resulting in singlet carbenes.''

'''''Electropositive''' substituents have a greater contribution to the HOMO, leading to high p character at the carbon centre and a small HOMO-LUMO gap, which favours electron promotion over spin pairing, resulting in triplet carbenes.''

However, it is important to note that these arguments only apply to substituents with valence p orbitals. The effects of d orbital participation has also not been considered, which may or may not be significant.

Sterically large substituents are proposed to alter the energy ordering of molecular orbitals and change the HOMO and LUMO to orbitals derived from degenerate orbitals of the substituent fragment orbitals. This decreases the HOMO-LUMO gap and favours the triplet. However, further calculations are required to support this proposal.

===References===
<references/>

Rep:Bc608 module3

2011-03-27T15:25:43Z

Bc608: /* Diels Alder Cycloaddition */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure 2'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure 6''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure 6''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure 7'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure 7:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure 8'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure 8''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure 9'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure 9''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure 10''' and the results are shown in '''Figure 11'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure 10''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure 11''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure 12'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure 12''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure 13'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure 13''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure 14'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure 14''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure 14'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure 15''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure 15:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure 15''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure 16'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure 16''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure 17'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure 17''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure 18'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure 18''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 19'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure 19''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure 19'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure 20'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure 20''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure 21'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure 21''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure 22'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure 22''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure 23'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure 23''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure 24'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure 24''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure 25'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure 25''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure 26''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure 27''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure 26''' and '''Figure 27''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure 28'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure 28''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure 28''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure 29'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure 29''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure 30''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure 31:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure 32'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure 32''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure 33'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure 33''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure 34'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure 34:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure 35'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}'''Figure 35:''' The calculated MOs of the starting materials at the DFT-B3LYP/6-31G(d) level of theory

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure 36'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure 36:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure 37'''.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure 37:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure 38'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure 38:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
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<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
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The IRC graphs are shown in '''Figure 39'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure 39:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure 40'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure 40:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}'''Figure 41:''' ''The calculated molecular orbitals of the transition state''

The HOMO of the transition state is comprised of contributions from the HOMO of ethylene and the HOMO-2 of cis-butadiene and is symmetric with respect to the plane.

The LUMO of the transition state is comprised of contributions from the HOMO of ethylene and the LUMO of cis-butadiene and is also symmetric with respect to the plane.

The HOMO-1 of the transition state is comprised of contributions from the LUMO of ethylene and the HOMO of cis-butadiene. It is antisymmetric with respect to the plane.

Orbital symmetry has therefore been preserved from reactants to the transition state. Molecular orbitals that were symmetric in the reactants must be symmetric in the transition state.

The difference between expected molecular orbitals in the transition state and the calculated molecular orbitals can be attributed to secondary orbital mixing, which was disregarded in the simple Frontier Molecular Orbital picture. This explains why the ethylene HOMO gives character to more than one bonding and antibonding set of molecular orbitals.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure 42''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure 43'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure 43''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed to give the energy of reactants. This value was then subtracted from the energy of the transition state. This calculation is valid because the transition state and the separate reactants only differ by their spatial separation. For butadiene, the energy of the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1 at 0 K.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure 44''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|} '''Figure 44:''' A table to show the results of the transition state optimisations

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure 45'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure 45:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
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</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
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</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure 46'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure 46:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure 47'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure 47:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure 48'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure 48:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure 49'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure 49:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}'''Figure 50:''' The dipole moment of the exo and endo transition states

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T15:18:27Z

Bc608: /* Optimising Chair and Boat structures */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure 2'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure 6''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure 6''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure 7'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure 7:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure 8'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure 8''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure 9'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure 9''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure 10''' and the results are shown in '''Figure 11'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure 10''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure 11''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure 12'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure 12''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure 13'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure 13''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure 14'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure 14''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure 14'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure 15''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure 15:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure 15''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure 16'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure 16''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure 17'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure 17''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure 18'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure 18''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 19'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure 19''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure 19'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure 20'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure 20''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure 21'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure 21''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure 22'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure 22''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure 23'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure 23''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure 24'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure 24''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure 25'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure 25''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure 26''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure 27''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure 26''' and '''Figure 27''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure 28'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure 28''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure 28''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure 29'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure 29''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure 30''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO of the transition state is comprised of contributions from the HOMO of ethylene and the HOMO-2 of cis-butadiene and is symmetric with respect to the plane.

The LUMO of the transition state is comprised of contributions from the HOMO of ethylene and the LUMO of cis-butadiene and is also symmetric with respect to the plane.

The HOMO-1 of the transition state is comprised of contributions from the LUMO of ethylene and the HOMO of cis-butadiene. It is antisymmetric with respect to the plane.

Orbital symmetry has therefore been preserved from reactants to the transition state. Molecular orbitals that were symmetric in the reactants must be symmetric in the transition state.

The difference between expected molecular orbitals in the transition state and the calculated molecular orbitals can be attributed to secondary orbital mixing, which was disregarded in the simple Frontier Molecular Orbital picture. This explains why the ethylene HOMO gives character to more than one bonding and antibonding set of molecular orbitals.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed to give the energy of reactants. This value was then subtracted from the energy of the transition state. This calculation is valid because the transition state and the separate reactants only differ by their spatial separation. For butadiene, the energy of the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1 at 0 K.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T15:12:44Z

Bc608: /* Energy minima on the C6H10 potential energy surface */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure 2'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure 6''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure 6''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure 7'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure 7:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure 8'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure 8''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure 9'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure 9''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure 10''' and the results are shown in '''Figure 11'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure 10''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure 11''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure 12'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure 12''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO of the transition state is comprised of contributions from the HOMO of ethylene and the HOMO-2 of cis-butadiene and is symmetric with respect to the plane.

The LUMO of the transition state is comprised of contributions from the HOMO of ethylene and the LUMO of cis-butadiene and is also symmetric with respect to the plane.

The HOMO-1 of the transition state is comprised of contributions from the LUMO of ethylene and the HOMO of cis-butadiene. It is antisymmetric with respect to the plane.

Orbital symmetry has therefore been preserved from reactants to the transition state. Molecular orbitals that were symmetric in the reactants must be symmetric in the transition state.

The difference between expected molecular orbitals in the transition state and the calculated molecular orbitals can be attributed to secondary orbital mixing, which was disregarded in the simple Frontier Molecular Orbital picture. This explains why the ethylene HOMO gives character to more than one bonding and antibonding set of molecular orbitals.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed to give the energy of reactants. This value was then subtracted from the energy of the transition state. This calculation is valid because the transition state and the separate reactants only differ by their spatial separation. For butadiene, the energy of the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1 at 0 K.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T15:09:25Z

Bc608: /* Activation Energy */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO of the transition state is comprised of contributions from the HOMO of ethylene and the HOMO-2 of cis-butadiene and is symmetric with respect to the plane.

The LUMO of the transition state is comprised of contributions from the HOMO of ethylene and the LUMO of cis-butadiene and is also symmetric with respect to the plane.

The HOMO-1 of the transition state is comprised of contributions from the LUMO of ethylene and the HOMO of cis-butadiene. It is antisymmetric with respect to the plane.

Orbital symmetry has therefore been preserved from reactants to the transition state. Molecular orbitals that were symmetric in the reactants must be symmetric in the transition state.

The difference between expected molecular orbitals in the transition state and the calculated molecular orbitals can be attributed to secondary orbital mixing, which was disregarded in the simple Frontier Molecular Orbital picture. This explains why the ethylene HOMO gives character to more than one bonding and antibonding set of molecular orbitals.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed to give the energy of reactants. This value was then subtracted from the energy of the transition state. This calculation is valid because the transition state and the separate reactants only differ by their spatial separation. For butadiene, the energy of the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1 at 0 K.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T15:03:00Z

Bc608: /* Molecular Orbitals of the Transition State */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO of the transition state is comprised of contributions from the HOMO of ethylene and the HOMO-2 of cis-butadiene and is symmetric with respect to the plane.

The LUMO of the transition state is comprised of contributions from the HOMO of ethylene and the LUMO of cis-butadiene and is also symmetric with respect to the plane.

The HOMO-1 of the transition state is comprised of contributions from the LUMO of ethylene and the HOMO of cis-butadiene. It is antisymmetric with respect to the plane.

Orbital symmetry has therefore been preserved from reactants to the transition state. Molecular orbitals that were symmetric in the reactants must be symmetric in the transition state.

The difference between expected molecular orbitals in the transition state and the calculated molecular orbitals can be attributed to secondary orbital mixing, which was disregarded in the simple Frontier Molecular Orbital picture. This explains why the ethylene HOMO gives character to more than one bonding and antibonding set of molecular orbitals.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:57:34Z

Bc608: /* Molecular Orbitals of the Transition State */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO of the transition state is comprised of contributions from the HOMO of ethylene and the HOMO-2 of cis-butadiene and is symmetric with respect to the plane. Orbital symmetry has therefore been preserved from reactants to the transition state.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:55:56Z

Bc608: /* Molecular Orbitals of the Transition State */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO of the transition state is comprised of contributions from the HOMO of ethylene and the HOMO of cis-butadiene and is symmetric with respect to the plane.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:54:41Z

Bc608: /* Molecular Orbitals of the Transition State */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO of the transition state is comprised of contributions from the HOMO of ethylene and the HOMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:52:20Z

Bc608: /* Molecular Orbitals of the Transition State */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of contributions from the HOMO of ethylene and the HOMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:45:54Z

Bc608: /* Transition State Optimisation */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

There is only one transition state of the Diels Alder reaction between ethylene and cis-butadiene. The σh plane of symmetry in both molecules means that attack from either the "top" face or "bottom" face is identical and thus endo and exo products do not exist. The transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
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<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
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The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:43:09Z

Bc608: /* IRC of the Diels Alder Transition State */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two following minima: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:41:54Z

Bc608: /* Transition State Optimisation */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout bond formation i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:40:47Z

Bc608: /* Transition State Optimisation */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:38:56Z

Bc608: /* Molecular Orbitals of Starting Materials */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric..<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> Molecular symmetry results in orbital symmetry. This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown below:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
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<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
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<jmol>
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<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
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====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
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<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
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|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
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||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
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|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
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|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
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<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
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<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
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IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
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<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
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<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
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====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:37:37Z

Bc608: /* Molecular Orbitals of Starting Materials */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactions are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric.<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown below:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:32:35Z

Bc608: /* Optimisation of starting materials */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. However, this is not unexpected. One would expect cis-butadiene to be a maxima on the PES, one would also expect it to be the reactive conformation of the Diels Alder reaction. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactants are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric.<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown below:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:29:39Z

Bc608: /* Optimisation of starting materials */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 90; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond.

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactants are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric.<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown below:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:29:02Z

Bc608: /* Optimisation of starting materials */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_AM1_FREQ.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The imaginary frequency</text>
</jmolAppletButton>
</jmol>

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond.

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactants are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric.<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown below:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
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</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
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|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
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</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:27:08Z

Bc608: /* The Diels Alder reaction between Ethylene and cis-butadiene */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
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<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
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|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
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<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition that proceeds via a concerted mechanism.

This reaction will be investigated using methods introduced in the previous exercise. The same general approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond.

The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain. The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond.

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactants are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric.<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown below:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:22:52Z

Bc608: /* Activation energies for the chair and boat transition states */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

===Conclusions===

The chair transition state is the preferred transition state of the Cope rearrangement of 1,5-hexadiene. The chair TS is 34 KJ mol-1 more stable than the boat TS at 298.15 K.

The methods of obtaining structures of transition states and information about what minima they connect have been used and understood.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition.

The same approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond, which would be freely rotating at room temperature. The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain.

The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond.

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactants are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric.<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
</jmolAppletButton>
</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown below:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
</jmolAppletButton>
</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
</jmolAppletButton>
</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
</jmolAppletButton>
</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
</jmolAppletButton>
</jmol>

IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
</jmolAppletButton>
</jmol> to the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
</jmolAppletButton>
</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>

Rep:Bc608 module3

2011-03-27T14:18:47Z

Bc608: /* Activation energies for the chair and boat transition states */

==Ben Chappell (CID: 00513494)==

==Introduction==

In this final module of the third year computational lab, transition state modelling is addressed. New computational techniques are utilised and an attempt has been made to understand each of these new techniques in the first section where they are introduced.

The first section is a worked tutorial investigating the Cope rearrangement of 1,5-hexadiene. Extensive references are made to "Appendix 1" throughout the section, which can be found here: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1

The second section investigates the two Diels Alder reactions.

As a side note: An extension was arranged with Dr Hunt for this wiki to be handed in at 5PM on the 26th of March 2011. Unfortunately, the whole Chemistry Wiki went down from around 5PM on the 25th of March 2011 and so this was not possible. Please email me (bc608@imperial.ac.uk) if there are any queries regarding late submission.

==Cope Rearrangement Tutorial==

Historically, the mechanism of the Cope Rearrangement <ref name="Cope Rearrangement">
W. von E. Doering and W.R. Roth, ''Angewandte Chemie'' 1963, '''75''', 27 </ref> was subject to much controversy. After numerous experimental and computational studies, the reaction is widely accepted to be concerted and proceed via either a chair or boat transition structure as shown in '''Figure 1'''<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

In this tutorial, a computational study is performed on the Cope Rearrangement of 1,5 Hexadiene as an example of how to approach a chemical reactivity problem.

[[Image:Bc608_CopeRearrangement.png |thumb|centre|500px|'''''Figure 1:''''' ''Cope rearrangement and retro-cope rearrangement of 1,5-hexadiene'']]

The aim of this exercise is to locate the low-energy minima and transition structure on the C6H10 potential energy surface, in order to determine the preferred reaction mechanism.

The approach that will be taken is as follows:

* Locate the low-energy minima on the C6H10 potential energy surface using a computationally inexpensive method
* Locate the high-energy maxima transition state structures on the C6H10 potential energy surface using a computationally inexpensive method
* Find which reactants and products these transition state structures link using a computationally inexpensive method
* Reoptimise structures of interest at a higher level of theory to provide an improved agreement with experiment
* Demonstrate that this approach is valid
* Use the above information in order to determine the preferred reaction mechanism

===Energy minima on the C6H10 potential energy surface===

Before calculations are carried out, it's important to firsty try and predict what the minima of the C6H10 potential energy surface may look like. This will allow the reasonability of the calculated minimas to be judged and prevents the results of calculations being followed blindly.

If we consider rotation about the central C-C bond of 1,5-hexadiene and assume that all interactions between groups are purely the result of steric hinderance, we can sketch a dihedral angle plot as shown in '''Figure X'''.

[[Image:Bc608_DihedralAngleOver360.png|thumb|centre|600px|'''Figure 2''' ''A sketched dihedral angle plot for rotation about the central C-C bond in 1,5-hexadiene'']]

Minima on the potential energy surface can be seen at 60° and 180° corresponding to the gauche (synclinal) and antiperiplanar conformations respectively. The anticlinal (120°) and synperiplanar (0°) conformations are maxima. It is likely therefore that stable conformations of 1,5-hexadiene will adopt a gauche ('''Figure 3''') or antiperiplanar ('''Figure 4''') relationship of the central 4 carbons. This then leaves only the terminal two carbons to be considered, which on first consideration might be expected to be rotated such that steric repulsions are minimised. The conformation of this central bond will therefore dominate the conformation of the rest of the molecule.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_GAUCHE.png|thumb|left|200px|'''Figure 3''' ''Gauche 1,5-hexadiene'']]
|
|[[Image:Bc608_APP.png|thumb|left|200px|'''Figure 4:''' ''Antiperiplanar 1,5-hexadiene complex'']]
|}

====Gauche versus antiperiplanar: Which is more stable?====

Now it has been predicted that the minima will either be "gauche" or "antiperiplanar", it's important to predict which one of the two will be more stable. Normally, calculated activation energies are referenced with respect to the lowest energy conformation of a reactant molecule.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> Complex reactant molecules may possess numerous possible conformations. A prediction of which conformer is most stable can allow the results of calculations to be assessed for plausibility.

It is proposed that 3 effects will determine the stability of the conformers of 1,5-hexadiene. These effects are:

# Sterics
# Stereoelectronics
# Van der Waals interactions

The effects of sterics have been discussed in the previous section and will not be discussed further. It is proposed that a small stereoelectronic effect may exist in 1,5-hexadiene, which may favour Gauche conformers.

Kirby's theory<ref name="Kirby">
A.J. Kirby in ''Stereoelectronic Effects (Oxford Chemistry Primers)'', Oxford University Press, 1996</ref> states:

''“There is a stereoelectronic preference for conformations in which the best donor lone pair or bond is anti-periplanar to the best acceptor bond or orbital”''

The "best" donor orbital is defined as the highest energy donor orbital. The "best" acceptor orbital is defined as the lowest energy acceptor orbital. A high energy donor orbital and a low energy acceptor orbital will minimise the interaction energy between the orbitals. It can be seen from the simplified form of the Klopman-Salem equation below that this will maximise the stabilisation energy imparted on the system.

[[Image:Klopman-salem.png|centre|200px]] [[Image:Bc608_StereoelectronicsGAUCHE.png|thumb|right|200px|'''Figure 5:''' ''Favourable orbital overlap in the Gauche conformer of the 1,5-hexadiene complex'']]

Where:

'''S2''' is the overlap integral

'''ESTAB''' is the stabilisation energy

'''ΔEi''' is the interaction energy or the energy difference between interacting orbitals

The highest energy donor orbital in 1,5-hexadiene is the σC-C and the lowest energy acceptor orbital is the σ*C-H orbital.<ref name="orbitalenergies">I.V. Alabugin and T.A. Zeidan, ''Journal of the American Chemical Society'' 2002, '''124''', 3175 - 3185 {{DOI|10.1021/ja012633z}}</ref> There will therefore be a stereo-electronic preference for C-H bonds and C-C bonds to be in an antiperiplanar arrangement, which is the case in Gauche conformers but not the case in antiperiplanar conformers. Thus stereoelectronics may favour the gauche conformation. This proposed favourable stereoelectronic interaction is shown in '''Figure 5'''.

The second effect to be considered is Van der Waals interaction. A larger number of attractive Van der Waals interactions between hydrogens may exist in the Gauche conformer, analogous to the case seen for n-butane. <ref name="Rzepa">H. Rzepa, ''Imperial College London Lecture Course Materials, Year 2: Conformational Analysis'' 2010, http://www.ch.ic.ac.uk/local/organic/conf/ accessed 22/03/11</ref> This so called "Gauche-effect" is especially abundant in large alkane chains, where "hairpin" structures are formed via Gauche linkages in order to maximise attractive Van der Waals forces. <ref name="gauche">L.L. Thomas, T.J. Christakis and W.L. Jorgensen, ''Journal of Physical Chemistry B'' 2006, '''110''', 21198–21204 {{DOI|10.1021/jp064811m}}</ref>

When the magnitude of these 3 effects is considered for the case of 1,5-hexadiene, one might expect both the Stereoelectronic and Van der Waals contributions to be small. Hydrogen and Carbon are very close in electronegativity and therefore σC-H and σC-C would be expected to be close in energy. σ*C-H and σ*C-C would also be expected to be close in energy. Therefore, the preference for σC-H being antiperiplanar to σ*C-C would probably be very weak. The stereoelectronic stablisation of the gauche conformers over the antiperiplanar conformers would be expected to be very small.

Van der Waals interactions are also probably insignificant. Van der Waals interactions themselves are weak and they are often only significant where they are numerous, additive interactions, as is the case in the hairpin structures described above. If the central C-C bond of 1,5-hexadiene is considered to be an ethylene unit, then one might expect Van der Waals interactions to exist between two chains each of a length of two carbons.

Due to the negligible contribution of these two effects, sterics would be expected to be the sole factor that determines conformer stability. One would therefore expect the most stable conformer of 1,5-hexadiene to have the central 4 carbons in an antiperiplanar arrangement.

====Locating the energy minima computationally====

In order to test the above hypothesis, two molecules of 1,5-hexadiene were drawn using Gaussview. One was drawn with an antiperiplanar relationship between the central four carbons and the other was drawn with a gauche relationship between the central four carbons. Whilst numerous gauche and antiperiplanar conformers may exist due to rotations about terminal carbons, the steric argument presented in the previous section should favour an antiperiplanar conformer significantly over any gauche conformer.

The two drawn structures were optimised with a Hatree Fock calculation using a 3-21G basis set. '''Figure X''' shows the results of these two calculations.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Conformation of the central 4 carbon atoms'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''||'''Appendix 1 Assignment
|-
| Antiperiplanar ||-231.691|| https://wiki.ch.ic.ac.uk/wiki/images/1/13/Bc608_APP_LINKAGE_1_5_HEXADIENE.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_APP_LINKAGE_1_5_HEXADIENE.mol</uploadedFileContents>
<text>APP</text>
</jmolAppletButton>
</jmol>
|| Anti4
|-
| Gauche ||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/c/c7/Bc608_GAUCHE3.LOG|| C1 ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Gauche3.mol</uploadedFileContents>
<text>Gauche</text>
</jmolAppletButton>
</jmol>
|| Gauche3
|}'''Figure X''': ''The optimised structures of two structures calculated using HF/3-21G''

The modelled Gauche conformer was found to be 4.4KJ mol-1 more stable than the modelled antiperiplanar conformer. Clearly there is a significant effect operating other than sterics. In order to see if there was a molecular orbital effect operating, the molecular orbitals of the two conformers were examined.

The HOMOs of the two modelled structures are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: center;"
|-
| rowspan="1"|
| colspan="4" |
|-
| [[Image:Bc608_HOMOOfMostStableGauche.png|thumb|left|200px|Gauche]]
|
|[[Image:Bc608_HOMOofapp.png|thumb|left|200px|Antiperiplanar]]
|}'''Figure X:''' ''The HOMOs for the modelled gauche and antiperiplanar conformers''

It can be seen that there is significant constructive secondary orbital overlap of π orbitals in the Gauche conformer, whereas this secondary orbital overlap is absent in the antiperiplanar conformer. One might expect different Gauche conformers to exhibit this secondary orbital overlap effect to different extents, dependent on the relative geometry of alkene groups. Antiperiplanar conformations however might be expected to be incapable of this secondary orbital overlap, due to the large distance between π orbitals in the structures.

Some groups have also postulated an attractive interaction between an alkene π orbital and a vinyl proton of the other alkene substituent<ref name="CH_PI">B.W. Gung, Z. Zhu and R.A. Fouch, ''Journal of the American Chemical Society'' 1995, '''117''', 1783-1788 {{DOI|10.1021/ja00111a016}}</ref> as shown in '''Figure X'''. However, no evidence of this effect could be found at this level of theory.

[[Image:Bc608_CH_PI_Interaction.png|thumb|centre|350px|'''Figure X''' ''A CH-π interaction proposed to account for the stability of Gauche conformers'']]

In conclusion, the lowest energy conformation of 1,5-hexadiene will either be the antiperiplanar conformation in which sterics are minimised or the gauche conformer in which constructive π overlap is maximised.

Consultation of Appendix 1<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> shows that gauche3, in which π orbital interactions are maximised
is indeed the most stable conformation of 1,5-hexadiene.

====Geometry optimisation the anti2 structure====

Despite it having previously been stated that calculated activation energies and enthalpies often use the lowest energy conformation of a reactant molecule as a reference, in this exercise the anti2 structure<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref> will be used as a reference. It's possible that this conformation is the most stable conformation at higher levels of theory or that the decision is just arbitrary.

The anti2 structure was drawn and optimised using a HF/3-21G calculation. The energy was found to match Appendix 1. The optimised structure was then reoptimised using a DFT-B3LYP/6-31G(d) optimisation. The results of the optimisations are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Calculation Type/Basis set'''|| '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Point group'''|| '''Structure'''
|-
| HF/3-21G||-231.693|| https://wiki.ch.ic.ac.uk/wiki/images/8/8b/Bc608_ANTI2_CONFORMATION.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ANTI2_CONFORMATION.mol</uploadedFileContents>
<text>HF/3-21G</text>
</jmolAppletButton>
</jmol>
|-
| DFT-B3LYP/6-31G(d) ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG|| Ci||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>DFT-B3LYP/6-31G(d)</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of anti 2 using HF/3-21G and DFT-B3LYP/6-31G(d) calculations''

It is important at this point to check that there is not a large difference between structures at the lower level of theory and the higher level of theory. Recall the approach that was being taken in this exercise: the potential energy surface was being mapped by a low level of theory (HF/3-21G), then points of interests are being reoptimised at a higher level of theory (DFT-B3LYP/6-31G(d)) in order to make comparisons to experiment. If there is little correlation between structures at the low level of theory and the higher level of theory, then there is a problem with this approach. Namely, the potential energy surface at the higher level of theory does not resemble the potential energy surface the the lower level of theory. Transition states and minima in the lower level of theory may not be transition states or minima at the higher level of theory. Thus it is imperative that a comparison between the two levels of theory is made to determine the degree to which structures change. A good agreement in parameters such as bond lengths and angles will mean that information obtained at the low level of theory can provide information about what is happening at the higher level of theory. A poor agreement will mean that the approach has to be revised. The referencing system for the comparisons is shown in '''Figure X''' and the results are shown in '''Figure X'''.

[[Image:Bc608_Referencingsystem.png|thumb|centre|400px|'''Figure X''' ''The referencing system used to compare dihedral angles'']]

{| class="wikitable" border="1" style="text-align: centre;"
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''Dihedral one angle''' || -114.7 || 118.5
|-
| '''Dihedral two angle''' || 180.0 || 180.0
|-
| '''Dihedral three angle''' || 114.7 || 118.5
|-
| '''H1-C-C-H3 dihedral angle''' || -62.8 || - 64.8
|-
| '''H1-C-C-H4 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H3 dihedral angle''' || 180.0 || 180.0
|-
| '''H2-C-C-H4 dihedral angle''' || 62.8 || 64.1
|-
| '''H-C-H bond angle'''|| 107.7 || 106.6
|-
| '''C=C bond length''' || 1.32 || 1.33
|-
| '''C-C bond length''' || 1.51 and 1.55 || 1.50 and 1.55
|-
| '''C-H bond length''' || 1.07 to 1.09 || 1.09 to 1.11
|}'''Figure X''': ''A table to compare dihedral angles between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Fortunately, the overall geometry of the structure does not change drastically upon reoptimisation with DFT/6-31G(d). The positions of the terminal carbons are controlled by '''Dihedral 1''' and '''Dihedral 3''' and these dihedral angles change by around 2.0°. The angle between hydrogens on the central carbons has also contracted by 1.1°. The bond lengths also change by around the order of error. These changes however are small, thus calculations at the HF/3-21G level of theory are likely to be giving relevant information about the potential energy surface at the DFT-B3LYP/6-31G(d) level of theory, which has been reported to be in close agreement to reality.<ref name="Phys3">
M. Bearpark, ''Imperial College Computational Lab Materials'' 2011, https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3 accessed 22/03/11 </ref>

====Frequency analysis of the optimised anti2 structure====

The energies discussed so far are meaningless in terms of experimental determinable quantities. They represent the energy of the molecule on the potential energy surface and do not take into account contributions such as zero point energy, thermal energy and entropy. In order to compare calculated energies to experimentally determined energies, these contributions must be taken into account. Fortunately, these contributions to the system energy can be obtained by a frequency calculation, which performs a thermochemical analysis as part of the calculation.

A frequency calculation was performed on the DFT-B3LYP/6-31G(d) optimised structure of anti2. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/f/fe/Bc608_DFT_FREQ_ANTICONF.LOG

The calculated frequencies were seen to be all positive and no imaginary frequencies were seen, confirming a minima had been reached. The calculated infra-red spectrum is shown in '''Figure X'''.

[[Image:Bc608_IR_spectrum_of_anti2_conf.png|thumb|centre|600px|'''Figure X''' ''The calculated IR spectrum of anti2 conformation of 1,5-hexadiene'']]

The Thermochemistry section of the output file was examined, 4 values were recorded. These are:

i) '''The sum of electronic and zero-point energies''' is the potential energy at 0 K including the zero-point energy and is given by the equation E = Eelec + ZPE

ii) '''The sum of electronic and thermal energies''' is the potential energy at 298.15 K and 1 atm, which is given by the equation E' = E + Evib + Erot + Etrans.

iii) '''The sum of electronic and thermal enthalpies''' is the previous energy, but corrected for room temperature it is given by the equation H = E' + RT

iv) '''The sum of electronic and thermal free energies''' includes entropic contributions and is given by the equation G = H - TS.

At 298.15 K, these values were calculated to be:

i) = -234.469 ± 3.808x10-3 Hartree

ii)= -234.462 ± 3.808x10-3 Hartree

iii) = -234.461 ± 3.808x10-3 Hartree

iv) = -234.501 ± 3.808x10-3 Hartree

An attempt was then made to calculate these values at 0.0 K. Examination of the log file however showed the calculation to still being performed at 298.15 K. It's possible that this is a bug where Gaussian is interpreting "0" as "default" rather than "0 K". The calculation was repeated with the temperature set as "0.0001 K". The log file is shown here: https://wiki.ch.ic.ac.uk/wiki/images/e/e8/Bc608_DFT_FREQ_ANTICONF_0K.LOG

i) -234.469 ± 3.808x10-3 Hartree

ii) -234.469 ± 3.808x10-3 Hartree

iii) -234.469 ± 3.808x10-3 Hartree

iv) -234.469 ± 3.808x10-3 Hartree

As expected, all values match the "sum of electronic and zero-point energies" value at 298.15 K.

==Optimising Chair and Boat structures==

Now the reactants and products of the Cope rearrangement have been optimised and their energies determined, it's now important to model the transition states of the reaction. The lowest activation energy pathway will give the fastest reaction. The fastest reaction will dominate the observed mechanism of the reaction.

There are a number of ways of modelling the transition state. This section of this report explores 3 complementary ways of achieving this.

These methods are:

* Optimise to Berny transition state directly
* Frozen co-ordinate preoptimisation, then optimise to Berny Transition State
* The QTS2 method.

The first two methods will be used to model the chair transition state and the last will be used to model the Boat Transition state. Discussion of how each method works is provided in each section.

===Direct optimisation to a Berny transition state (Chair TS)===

====How the method works====

In determination of the normal vibrational modes, the nuclear Schrödinger equation is solved. The nuclear Schrödinger equation is shown below:

[[Image:Bc608_NuclearSchrodinger.png|centre|200px|]]

The potential energy term, V(x) can be calculated using a Taylor expansion about x=0. This gives the following equation:

[[Image:Bc608 TaylorExpansion2.png|centre|500px|]]

Where dxi and dxj are displacements of degrees of freedom. Examples of degrees of freedom include: translation in the x direction of atom 1, translation in the y direction of atom 1, translation of atom 2 in the z direction etc. Assuming the Harmonic Oscillator approximation to apply:

# Terms higher than x2 are assumed to be negligible.
# The x term is zero as the gradient at a stationary point is zero.
# V(0) is equal to zero.

This simplifies the Taylor expansion to:

[[Image:Bc608_SimplifiedTaylorExpansion2.png|centre|300px|]]

This resembles the Simple Harmonic Oscillator equation, where the second derivative term resembles the spring constant. However, there is a mass dependence in the spring constant of a simple harmonic oscillator, whereas there is no mass dependence here. We cannot equate the two yet, but they are clearly related. This mass dependence will be accounted for shortly. The terms of the sum of second derivatives can be projected out in a matrix form as shown below.

[[Image:Bc608 DiagonalisationOfHessian.png|centre|300px|]]

This is the '''Cartesian Hessian''' containing the internal degrees of freedom of the system. The mass dependence of vibrations can now be introduced by converting the cartesian coordinates into mass-weighted-coordinates. Mass-weighted-coordinates are related to cartesian coordinates according to the following equation:

[[Image:Bc608_MAsscoords.png|centre|200px|]]

Where qi is a mass-weighted-coordinate, xi is a cartesian coordinate and mi is the mass of atom i.

The Cartesian Hessian can then be converted into the mass-weighted-coordinate Hessian and equated to the force constant:

[[Image:Bc608_HessianConversion.png|centre|200px|]]

The mass-weighted-coordinate Hessian can now be put back into the nuclear Schrödinger equation and solved. In order to solve the nuclear Schrödinger equation, the Hessian must be diagonalised to give a diagonal matrix of eigenvalues. The matrix of normal modes is an orthogonal matrix and so the inverse of the matrix is equal to the transposed matrix.

[[Image:Bc608_Schrodinger.png|centre|200px|]]

Where X is the matrix of normal modes, H is the mass-weighted-coordinate Hessian matrix and α is the diagonal eigenvalue matrix. The square root of α gives the frequency of a normal mode. The Hessian is diagonalised into a matrix of eigenvalues because normal modes are orthogonal to one another. When the transposed normal mode matrix (X+) is post multiplied by HX, any off diagonal terms are equal to zero as there is no overlap between different normal modes. When an element of the eigenvalue matrix is negative, this gives an imaginary frequency as the square root of a negative number is an imaginary number.

The first step of the optimise to Berny transition state method computes the mass-weighted-coordinate Hessian matrix and thus the normal modes of the system. The imaginary normal mode is identified by finding the normal mode that corresponds to the negative eigenvalue in the diagonal matrix of eigenvalues. The TS optimisation then moves nuclei in a manner which increases the energy of the degree of freedom that corresponds to the imaginary mode (the reaction coordinate), whilst minimising the energy of all other degrees of freedom. The Hessian is then updated using an algorithm. An algorithm is used to update the Hessian as its "from scratch" calculation is computationally expensive - calculating the Hessian at iteration would therefore lead to very slow calculations. The procedure of atom position adjustment and updating the Hessian is then repeated iteratively until the reaction coordinate is at a maxima and all other degrees of freedom are at a minima. This iterative procedure is illustrated in '''Figure X'''.

[[Image:Bc608 BernyTShowitworks.png|thumb|centre|900px|'''Figure X''' ''A cartoon to show the Berny optimisation procedure. The left hand diagram shows the position of the initial starting geometry on the 3 potential energy surfaces corresponding to the normal modes. The right hand diagram shows the position of the transition state on the 3 potential energy surfaces.'']]

The optimisation to the transition state is an iterative procedure and the Hessian is updated using an algorithm at each iteration. The Hessian therefore becomes less accurate as the iterations proceed, so there will be a limit to the number of iterative steps that can be taken. Furthermore, the optimisation procedure assumes that the potential energy surface has a quadratic shape - this assumption is only valid when the system is close to the transition state geometry. The method will only work as long as the quadratic potential energy surface approximation holds.

====Applying the method====

The chair transition state consists of two C3H5 allyl fragments positioned approximately 2.2Å apart with C2H symmetry. The chair transition state was shown in '''Figure X'''. In order to model this transition state, this C3H5 allyl fragment must first be optimised.

[[Image:Bc608_CHAIR_TS.png|thumb|centre|400px|'''Figure X''' ''The chair transition state structure'']]

The C3H5 allyl fragment was optimised using a HF/3-21G calculation. The energy of the molecule was found to be -115.823 ± 3.808x10-3 Hartree and the point group was found to be CS. The log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/6/68/Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.LOG

The optimised structure can be viewed here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ALLYL_FRAGMENT_GEO_OPT_HF_3_21_G.mol</uploadedFileContents>
<text>Allyl Fragment Jmol</text>
</jmolAppletButton>
</jmol>

A guessed structure of the chair transition state was then constructed. Two allyl fragments were pasted into a new window and orientated so that the guessed structure resembled the chair transition state shown in '''Figure X'''. The two allyl fragments that come together in the reaction were set roughly 2.2Å apart. The guessed structure can be viewed here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_Chair_TS_Guess.mol</uploadedFileContents>
<text>Guessed Chair Transition State</text>
</jmolAppletButton>
</jmol>

The guessed transition state structure was pasted into a new window. An optimisation with frequency analysis to a Berny Transition State was performed using a Hatree Fock calculation and a 3-21G basis set. The Hessian was calculated at the start and updated throughout the calculation. The additional keywords "Opt=NoEigen" were included to prevent the calculation crashing if more than one imaginary frequency is detected during the optimisation procedure. In the case of detection of multiple negative frequencies, the most negative frequency is deemed to be the reaction co-ordinate. A log file of the calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/1/1a/Bc608_BERNY_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised structure of the Berny Transition State is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Berny_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Berny Chair Transition State</text>
</jmolAppletButton>
</jmol>

One imaginary frequency was seen at -818cm-1, which corresponds to the reaction co-ordinate. This vibration shows the Cope Rearrangment via a chair transition state. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_VIBRATIONS_METHODONE.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>The reaction co-ordinate</text>
</jmolAppletButton>
</jmol>

===Indirect optimisation to a Berny transition state: the frozen coordinate method (Chair TS)===

====How the method works====

The frozen coordinate method is very closely related to the direct optimisation to a Berny transition state. In this method however, there is a prior preoptimisation, which attempts to bring all degrees of freedom other than the reaction coordinate to a minima. '''Figure X''' shows the approach of the frozen coordinate method compared to the approach of the direct optimisation to a Berny transition state.

[[Image:Bc608 FrozenCoordinateMethodvsdirect.png|thumb|centre|600px|'''Figure X:''' A 2D PES to compare and contrast the direct optimisation to Berny TS method and the frozen coordinate method]]

In the preoptimisation, bond forming/breaking distances are fixed. This freezes the degree of freedom corresponding to the reaction coordinate and prevents it from changing. An optimisation is then performed to attempt to minimise the energy of all other degrees of freedom. This is represented in '''Figure X''' by the orange arrow.

It is important to note that the other degrees of freedom may involve translations or rotations of the atoms involved in bond forming/breaking. A consequence of this is that a minima with respect to each 'other degree of freedom' may not be reached. Reaching the minima in some of these degrees of freedom may require a change in motions of atoms involved in bond forming/breaking. In cases where the energy with respect to a degree of freedom cannot be minimised fully, the calculation makes the energy as close as possible to the minima.

This preoptimisation is then followed by an optimisation to a Berny transition state as before. The frozen coordinate method provides the Berny transition state calculation with atom positions that are much closer to the transition state atom positions, so the optimisation is much more likely to converge.

====Applying the method====

The guessed chair transition state structure was then optimised using the frozen co-ordinate method. The guessed TS structure was pasted into a new window. The reaction co-ordinate was frozen using the Redundant Co-ordinate Editor. Additional keywords "Opt=ModRedundant" were included and a Hatree Fock optimisation using a 3-21G basis set performed.

The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/0/07/Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG The jmol of the resulting structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The checkpoint file of the previous calculation was opened. The redundant co-ordinate editor was then used to calculate the derivative along the reaction co-ordinate. The force constants were not calculated. A Berny transition state optimisation was then performed, in which only the reaction co-ordinate was allowed to vary until a maxima was reached.

A log file for the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/8/8f/Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.LOG

The optimised transition state can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_REDUNDANT_COORDS_TS_OPTIMISATION_HF_3_21_G.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The vibration corresponding the the reaction co-ordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Method2_VIBRATIONS.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The bond forming and bond breaking lengths were measured to be 2.02Å ± 0.005 in both methods, showing that the methods are complementary and achieve the same optimised chair transition state.

===QTS2 Method (Boat transition state)===

====How the method works====

The QTS2 method relies on Rolle's theorem to reach a transition state. Rolle's theorem states:

''If two values of a function f(x) are equal at x = a and x = b, then there must exist a point c between a and b where the first derivative of the function is 0.''

This is shown pictorially in '''Figure X'''.

[[Image:Bc608_HowQST2works.png|thumb|centre|400px|'''Figure X''' ''A diagram to illustrate Rolle's theorem'']]

Applied to a chemical system, this means that if two minima are on the same potential energy surface and are of the same energy, with no other minima between them, then there must be a maxima that connects the two minima. This can be used to locate the transition state of a reaction, which is an energy maxima.

Atomic positions are altered so that a provided "reactant structure" is mapped onto a provided "product structure". Between these two structures will lies an intermediate structure of which the first derivative of energy with respect to atomic displacement is zero. This structure will correspond to the structure of the transition state.

====Applying the method====

The boat transition state structure is reached by the QST2 method. The boat transition state is shown in '''Figure x'''.

[[Image:Bc608_Boat_ts.png|thumb|centre|400px|'''Figure x''' ''The boat transition state structure'']]

The structures of reactant and product used the previously optimised anti2 structure and the atom numbers were altered as shown in '''Figure X'''.

[[Image:Bc608_ReactantsandproductsBoatTS.png|thumb|centre|500px|'''Figure X''' ''The reactants and products for the Cope rearrangement'']]

A HF/3-21G calculation was used to interpolate between these two structures to a QST2 transition state. The log file for the calculation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/0/02/Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.LOG

The job was found to fail, with the following structure reached:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_ONE.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The QST2 calculation does not consider the possibility of rotation about bonds - it is only capable of direct linear mapping of "reactant" onto "product". From this starting geometry, neither the boat nor chair transition structure can be reached linearly. The boat transition state would require prior rotation about the central C-C bond between carbons 3 and 4, which the calculation does not consider. A chair transition state cannot exist chemically between these structures, as this would require an unconcerted mechanism, in which the central bond broke before the new bond formed. The structure looks like a dissociated chair transition state, so it is possible that this is what has been attempted.

In order to achieve the boat transition state, the starting geometries had to be altered so that this prior rotation had already taken place before the QST2 calculation was attempted. The dihedral angle between C2-C3-C4-C5 was set as 0°. The C2-C3-C4 and C3-C4-C5 angles were then set as 100°. The new starting geometries are shown in '''Figure 12'''.

[[Image:Bc608_BOAT_REACTANTSANDPRODUCTSTS2.png|thumb|centre|400px|'''Figure X''' ''The new reactant and product structures for the Cope rearrangement'']]

The QST2 calculation was repeated with the starting geometries shown in '''Figure X'''. The log file of the calculation can be seen here: https://wiki.ch.ic.ac.uk/wiki/images/5/5e/Bc608_QST2_BOAT_OPT_ATTEMPT_TWO.LOG

The structure of the boat transition state is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_BOAT_OPT_ATTEMPT_TWO_AFTER_REJIG.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The one imaginary frequency corresponding to the reaction coordinate can be seen here:<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_QST2_VIBRATIOS_BOAT_OPT.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

===IRC===

Transition states connect minima (reactants and products) on a potential energy surface. Now the transition states have been optimised its important to find which minima they connect. This can be achieved by the Intrinsic Reaction Coordinate method.

====How the method works====

The IRC method follows the minimum energy path from a transition state structure down to its local minimum on a potential energy surface. In the IRC calculation, the Hessian matrix is calculated at the starting geometry and the reaction coordinate is recognised by the negative element of the diagonal eigenvalue matrix.

A small step (step size left as default) is then taken in the direction of the reaction coordinate in the direction where the gradient of the potential energy surface is steepest. In this exercise, the forward and backward directions are identical and thus only the forward direction was calculated. The reaction coordinate is then frozen and the atomic positions are changed so that all other degrees of freedom are optimised. This gives the minimum energy structure for the specified reaction coordinate value. Another small step is then taken in the direction of the reaction coordinate. The other degrees of freedom are then optimised. This process is repeated in an iterative procedure, until a stationary point is reached. Each iteration of this IRC procedure will be referred to as an '''IRC iteration''' from here-on-in. This is important to define, as the optimisation of all other degrees of freedom at each value of reaction coordinate is also an iterative process. Each iteration in the optimisation of all other degrees of freedom will be referred to as an '''Optimisation iteration'''. The IRC procedure is shown in '''Figure X'''.

[[Image:Bc608_IRC_CALC_WHATISHAPPENING.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the IRC procedure. Each orange spot is an IRC iteration.'']]

As mentioned, for each IRC iteration an optimisation of all non-reaction coordinate degrees of freedom is performed, requiring several optimisation steps. The optimisation procedure uses the inverse Hessian matrix. In the IRC method, the Hessian matrix can either be calculated for the very first IRC iteration and then updated throughout the IRC iterations by an algorithm. Alternatively, the Hessian matrix can be recalculated for every IRC iteration. In both cases, the Hessian is updated by an algorithm throughout the the optimisation procedure. The first method is named "Calculate Force Constants Once", whereas the second method is named "Calculate Force Constants Always". Whilst "Calculate Force Constants Once" gives much more rapid calculations, it can lead to a problem in optimising structures. In the method, the Hessian becomes increasingly less accurate with IRC iterations i.e. an introduced error in the Hessian at IRC iteration #3 will be present in IRC iteration #5. Furthermore, errors will be additive. "Calculate Force Constants Always" is computationally much more demanding, but the Hessian never becomes very inaccurate because it is thrown out and recalculated from scratch for every IRC iteration.

After a large number of IRC iterations, the inaccuracy introduced into the Hessian can cause the optimisation procedure to fail. A maximum number of 20 optimisation iterations are permitted on the version of Gaussian used by SCAN, whereas 58 are permitted on the version of Gaussian used by departmental computers. After the maximum number of optimisation iterations is reached, the optimisation procedure is abandoned. The effect of Hessian accuracy and therefore calculation method upon optimisation iterations is shown in '''Figure x'''.

[[Image:Bc608_IRC_PES_CALCULATIONS_FORCECONSTANTALLVSGUESS.png|thumb|centre|400px|'''Figure X''' ''A cartoon to show the effect of Hessian calculation method upon geometry optimisation'']]

====Applying the method (Chair TS)====

The .chk file of the chair transition state structure optimised by the Frozen co-ordinates method was opened in Gaussview. The IRC method was employed in a Hatree Fock calculation using a 3-21G basis set. The IRC was computed in the forward direction only because the IRC will be symmetrical as reactants and products are the same structure. The force constants were calculated only once at the start of the calculation. 50 points along the IRC were used. Unfortunately the log file was too large to be uploaded and the optimisation was found to fail.

To try and find out why the optimisation failed, the logfile was examined. It was found that the 42nd optimisation had not converged.

Item Value Threshold Pt 42 Converged?
Maximum Force 0.001092 0.000450 NO
RMS Force 0.000194 0.000300 YES
Maximum Displacement 0.002061 0.001800 NO
RMS Displacement 0.000446 0.001200 YES

58 attempts were made to make the 42nd IRC iteration converge to an optimum geometry, after all of these 58 attempts failed the optimisation was stopped as the maximum number of optimisation iterations had been reached.

The structure of the 41st IRC iteration, the last one to converge is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_ATTEMPT1_41stiteration.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_50pointsIRCgraphs.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

Increasing the number of IRC iterations will '''not''' help in this case. The calculation has run out of optimisation iterations, it has not run out of IRC iterations. In order to reach the local minimum, there are two approaches that could be taken.

# The 41st IRC iteration of the previous IRC calculation could be optimised to a minimum. This relies on the 41st IRC point of the previous calculation being sufficiently close in structure to the local minimum that the IRC was seeking. If the 41st IRC point is far away from the local minimum, a false minimum may be converged to.
# The IRC can be repeated with force constants calculated always. This will lead to the Hessian being more accurate throughout the 41st IRC iteration. The situation described in '''Figure X''' may apply in this case. This approach will be computationally more demanding than approach 1) however, but avoids the possibility of converging to a false minimum.

Both approaches were attempted, in order.

'''Direct optimisation of the 41st IRC iteration of the previous IRC calculation'''

A direct optimisation was performed on the 41st point of the previous IRC calculation. The log file of the optimisation is shown here: https://wiki.ch.ic.ac.uk/wiki/images/d/d5/Bc608_STRAIGHTOFFOPTMISATION.LOG

The optimised structure is shown here:<jmol><jmolAppletButton>
<uploadedFileContents>Bc608_StraightOffOptmisation.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''.

'''Repeated IRC with a force constants calculated at every iteration'''

The IRC calculation was repeated with 50 IRC steps and the Hessian being calculated at the start of every IRC iteration.

The log file of the calculation can be found here: {{DOI|10042/to-8012}} The IRC was found to complete successfully. The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_FORCEconstantsALWAYSProgressAlongIRC.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_ForceconstantsalwaysSCAN.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as ''Gauche2''. The chair transition state therefore connects 1,5-hexadienes in the ''Gauche2'' conformation.

====Applying the method (Boat TS)====

The IRC was also calculated for the boat transition state with the Hessian being recalculated at every IRC iteration. The log file of the calculation can be found here: {{DOI|10042/to-8013}} The IRC calculation was found to be successful.

The progress of the IRC calculation can be seen in '''Figure X'''.

[[Image:Bc608_Boat_TS_IRC_GRAPH.png|thumb|centre|800px|'''Figure X''' ''The progress of the IRC calculation'']]

The final structure can be found here: <jmol><jmolAppletButton>
<uploadedFileContents>Bc608_IRC_BOAT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>

This structure is referred to in Appendix 1 as '''Gauche3'''. The boat transition state therefore connects 1,5-hexadienes in the '''Gauche3''' conformation.

===Comparison between low level theory, high level theory and experiment===

As discussed in the introduction, the potential energy surface of C6H10 is mapped with a low level theory and then points of interest optimised at a higher level of theory so that comparisons to experiment can be made. Now the transition states have been optimised at a low level of theory, a reoptimisation at a higher level of theory must now be performed.

The previously optimised chair and boat transition state structures were reoptimised using DFT-B3LYP/6-31G(d) calculations. The results are shown in '''Figure x'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file''' || '''Jmol'''
|-
| Chair TS || -234.557 || {{DOI|10042/to-8015}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Chair_B3Lyp.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Boat TS || -234.543 || {{DOI|10042/to-8016}} || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BOAT_B3LYP.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Anti2 ||-234.612|| https://wiki.ch.ic.ac.uk/wiki/images/b/bc/Bc608_DFT_OPT_ANTICONF2.LOG||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_OPT_ANTICONF2.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The relevant optimised structures at the DFT-B3LYP/6-31G(d) level of theory''

====Comparison of geometrical parameters====

As discussed in the ''Geometry optimisation the anti2 structure'' section, the geometrical parameters of both the chair and boat transition state structures must be compared in order to show that the approach of mapping the PES with a low level theory is valid.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Chair Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 124.3 || 119.9
|-
| '''H-C-H''' || 117.4 || 112.5
|-
| '''H-C-C-H''' || 180.0 and 0.0 || 163.6 and 22.6
|-
| '''C-C''' || 1.39 || 1.41
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.06 and 2.12 || 1.97
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| colspan="4"|'''Boat Transition State'''
|-
| rowspan="2"|'''Parameter'''
| colspan="2"|'''Calculation type'''
|-
| '''HF/3-21G'''|| '''DFT-B3LYP'''
|-
| '''C-C-C''' || 121.7 || 122.3
|-
| '''H-C-H''' || 114.7 || 114.4
|-
| '''H-C-C-H''' || 167.0 and 17.4 || 167.5 and 18.2
|-
| '''C-C''' || 1.38 || 1.39
|-
| '''C-H''' || 1.07 || 1.09
|-
| '''Bond forming/breaking distance''' || 2.14 || 2.21
|}'''Figure X''': ''A table to compare geometrical parameters between the optimised structures from HF/3-21G and DFT-B3LYP/6-31G(d) calculations. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

As can be seen from '''Figure X''' and '''Figure X''', the geometrical parameters are very similar between the two levels of theory for both transition states. Therefore, the approach to this exercise is valid.

====Comparison of energy differences====

The energy differences relative to the ''anti2'' structure were then calculated. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculation type''' || '''Relative energy to ''anti2'' at the same level of theory ± 10 /KJ mol-1'''
|-
| ''Chair TS'' || HF/3-21G || 194
|-
| || DFT/6-31G(d) || 144
|-
| ''Boat TS'' || HF/3-21 || 236
|-
| || DFT/6-31G(d) || 181
|}'''Figure X''': ''A comparison of energies relative to the ''anti2'' structure at HF/3-21 and DFT/6-31G(d) levels of theory ''

As can be seen in '''Figure X''', there is a large difference in the relative energies between the two theories. This shows that the reoptimisation is required - whilst the geometries are similar, the energies of the systems are very different.

====Activation energies for the chair and boat transition states====

The activation energies for the reaction via the chair and boat transition states at 0 K are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 ''' || '''Literature<ref name="LitActivationenergy">
M.J. Goldstein and M.S. Benzon, ''Journal of the American Chemical Society'' 1972, ''94'', 7147 {{DOI|10.1021/ja00775a046}}</ref> Activation Energy /KJ mol-1'''
|-
| ''Chair TS'' || 144|| 140.2 ± 2.1
|-
| ''Boat TS'' || 181|| 187.0 ± 8.4
|}'''Figure X''': ''A table to compare the calculated activation energies to literature values''

The experimentally determined values are in excellent agreement with the calculated values. At 298.15 K these values were calculated as:

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Calculated Activation energy ± 10 /KJ mol-1 '''
|-
| ''Chair TS'' || 139
|-
| ''Boat TS'' || 173
|}'''Figure X''': ''A table to show the calculated activation energies at 298.15 K''

As expected, the mechanism via the chair transition state has the lowest activation energy. The chair TS avoids eclipsing interactions between groups that are present in the more strained boat TS. The activation energies are also reduced at 298.15 K, which reflects the energy supplied to the system by the surroundings.

==Diels Alder Cycloaddition==

===The Diels Alder reaction between Ethylene and cis-butadiene===

[[Image:Bc608_Butadiene+EthyleneRxn.png|thumb|250px|centre|'''Figure X:''' ''The Diels Alder reaction between Ethylene and cis-butadiene'']]

The Diels Alder reaction between ethylene and cis-butadiene is a 4πs + 2πs cycloaddition.

The same approach as the previous exercise will be taken. The potential energy surface will be mapped using low level (AM1) calculations and points of interest will be reoptimised using higher level (DFT-B3LYP/6-31G(d)) calculations.

====Optimisation of starting materials====

Ethylene and cis-butadiene were optimised using semi-empirical AM1 calculations. A frequency analysis was then performed to characterise the stationary points reached. The results are shown below in '''Figure X'''

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || 0.049 || https://wiki.ch.ic.ac.uk/wiki/images/2/27/Bc608_BUTADIENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/7/7b/Bc608_BUTADIENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Butadiene_AM1_opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene || 0.026 || https://wiki.ch.ic.ac.uk/wiki/images/f/f5/Bc608_ETHYLENE_AM1_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/9/98/Bc608_ETHYLENE_AM1_FREQ.LOG || <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Ethylene_AM1_Opt.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No negative frequencies were seen for the ethylene structure, confirming a minima had been reached. One negative frequency was seen however for the cis-butadiene at -36 cm-1. This frequency corresponds to rotation about the central C-C bond, which would be freely rotating at room temperature. The negative frequency suggests that this conformer is a maxima, as would be expected due to the synperiplanar arrangement of atoms leading to significant A1,4 strain.

The two structures were then reoptimised using DFT-B3LYP/6-31G(d) calculations and frequency analysis was performed. The results are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Structure''' || '''Total Energy ± 3.808x10-3 / Hatree'''|| '''Log file of optimisation'''||'''Log file of frequency analysis || '''Jmol'''
|-
| Cis-butadiene || -155.986|| https://wiki.ch.ic.ac.uk/wiki/images/c/c1/Bc608_BUTADIENE_B3LYP_OPT.LOG||https://wiki.ch.ic.ac.uk/wiki/images/a/aa/Bc608_BUTADIENE_B3LYP_FREQ.LOG‎|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BUTADIENE_B3LYP_OPT.mol</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|-
| Ethylene ||-78.587 || https://wiki.ch.ic.ac.uk/wiki/images/3/3b/Bc608_ETHYLENE_B3LYP_OPT.LOG ||https://wiki.ch.ic.ac.uk/wiki/images/8/86/Bc608_ETHYLENE_B3LYP_FREQ.LOG|| <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ETHYLENE_B3LYP_OPT.mol‎</uploadedFileContents>
<text>Jmol</text>
</jmolAppletButton>
</jmol>
|}'''Figure X''': ''The optimised structures of Cis-butadiene and Ethylene''

No imaginary frequencies were seen for ethylene and the cis-butadiene again showed one imaginary frequency corresponding to rotation about the central C-C bond.

====Molecular Orbitals of Starting Materials====

The Diels Alder reaction of cis-butadiene and ethylene is a pericyclic reaction, in which the π orbitals of ethylene (dieneophile) are used to form new σ bonds with the π orbitals of cis-butadiene (diene). Pericyclic reactants are governed by Woodward-Hoffman's rules<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref>. The rules state that a cycloaddition involving 6π electrons will proceed suprafacially under heat.

The nodal properties of a system can be used in order to predict whether a reaction is allowed or disallowed.

* If the HOMO of one reactant can interact with the LUMO of the other reactant then the reaction is allowed.
* The HOMO and LUMO can only interact when there is significant overlap between the orbitals. If the orbitals have different symmetry properties then no overlap is possible and the reaction is forbidden.

Both cis-butadiene and ethylene have a mirror plane, about which their molecular orbitals must be symmetric.<ref name="WoodwardHoffman"> R.B. Woodward and R. Hoffmann, ''Angewandte Chemie International Edition'' 1969, '''8''', 781-853 {{DOI|10.1002/anie.196907811}}</ref> This mirror plane is shown in '''Figure X'''.

[[Image:Bc608_MirrorPlane.png|200px|thumb|centre|'''Figure X:''' ''The mirror plane about which reactant MOs must be symmetric'']]

The calculated MOs of the starting materials are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure''' || '''HOMO''' || '''LUMO'''
|-
| Cis-butadiene || [[Image:Bc608_Butadiene_HOMO.png|200px|thumb| ''Antisymmetric with respect to the plane'']] || [[Image:Bc608_Butadiene_LUMO.png|200px|thumb|''Symmetric with respect to the plane'']]
|-
| Ethylene || [[Image:Bc608_Ethylene_HOMO.png|200px|thumb|''Symmetric with respect to the plane'']] || [[Image:Bc608_Ethylene_LUMO.png|200px|thumb|''Antisymmetric with respect to the plane'']]
|}

It can be seen that the frontier molecular orbitals of ethylene and cis-butadiene are complementary. The HOMO of ethylene can interact with the LUMO of cis-butadiene and vice versa as the orbitals are of the same symmetry. Therefore, one would predict the cycloaddition reaction between these two starting materials to be allowed. In order to test this prediction, the molecular orbitals of the transition state must be examined. Before this can be done however, the transition state must first be optimised.

====Transition State Optimisation====

The structure of the transition state of the cycloaddition between ethylene and cis-butadiene has an envelope like structure in order to maximise overlap of the π orbitals. A sketch of the transition structure is shown in '''Figure X'''.

[[Image:Bc608_EnvelopeTS.png|thumb|150px|centre|'''Figure X:''' ''A sketch of the transition state of the cycloaddition'']]

This guessed transition state structure was drawn on Gaussview by first drawing a norbornene like structure shown in '''Figure X'.

[[Image:Bc608_Norbornene_Start_Structure.png|thumb|150px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

This initial norbornene structure was then optimised using semi-empirical AM1 calculations. The log file for this calculation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/d/db/Bc608_NORBORNENE_PRECURSOR.LOG

The optimised structure can be seen here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_NORBORNENE_PRECURSOR.mol</uploadedFileContents>
<text>Norbornene Precursor</text>
</jmolAppletButton>
</jmol>

The bridging CH2-CH2 unit was then removed and the structure split into the two fragments that come together in the Diels Alder reaction. The distance between these two fragments was then set at 2.2Å. This process is shown in '''Figure X'''.

[[Image:Bc608 DrawingProcedureForPartB.png|thumb|400px|centre|'''Figure X:''' ''The starting structure used to draw the guessed transition state structure'']]

The starting geometry for the transition state optimisation is shown here: <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Starting_geometry.mol</uploadedFileContents>
<text>Starting Geometry for TS optimisation</text>
</jmolAppletButton>
</jmol>

An optimisation to Berny Transition State was then performed using an AM1 semi-empirical calculation. The log file for the calculation can be found here. https://wiki.ch.ic.ac.uk/wiki/images/7/73/Bc608_BERNY_TS_OPT.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_BERNY_TS_OPT.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
</jmolAppletButton>
</jmol>

One negative frequency at 956cm-1 was seen, which corresponded to the Diels Alder Reaction. <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_TS_FREQUENCY.LOG</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
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The structure of the transition state was then reoptimised at the DFT-B3LYP/6-31G(d) level of theory. A log file for the optimisation can be found here: https://wiki.ch.ic.ac.uk/wiki/images/a/a4/Bc608_DFT_TS_OPT_ETHYLENEANDBUTADIENE.LOG

The optimised transition state structure can be found here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_OPT_ethyleneandbutadiene.mol</uploadedFileContents>
<text>Optimised Transition State Structure</text>
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</jmol>

One negative frequency at 525 cm-1 was seen, which corresponded to the Diels Alder Reaction.
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 3; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Diels Alder reaction</text>
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</jmol>

It can be seen that the formation of both C-C σ bonds is synchronous and the reaction is a concerted process.

The lowest positive frequency was seen at 136 cm-1 and is shown below:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_DFT_TS_FREQ.out</uploadedFileContents>
<script>frame 4; vectors 4; rotate 180; vectors scale 4; color vectors white; vibration 4;</script>
<text>Lowest positive frequency</text>
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</jmol>

The lowest positive frequency shows an asynchronous vibration. A positive vibrational frequency implies that displacements in the directions of these normal modes is uphill in energy and therefore unfavourable. During an asynchronous vibration, orbital symmetry about the plane is broken. This implies that the minimum energy pathway from the reactants through the transition state to the products involves retention of orbital symmetry throughout i.e. the bond formation is completely synchronous.

====IRC of the Diels Alder Transition State====

Now a transition state has been found, it's now important to prove that this transition states connects the two expected minima. An AM1 IRC calculation was performed on the AM1 optimised transition state structure. Force constants were calculated at every step and the IRC was calculated forwards and backwards. {{DOI|10042/to-8128}}

The IRC showed that the transition state connected the two expected minima. The IRC graphs are shown in '''Figure X'''.

[[Image:Bc608 IRC FOR PARTB GRAPHS.png|thumb|600px|centre|'''Figure X:''' ''The IRC for the transition state structure. The reaction coordinate goes from left to right - reactants to product'']]

The structures or the reactants and product can be seen here:
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_SM.mol</uploadedFileContents>
<text>Reactants</text>
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</jmol>
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_PRODUCT.mol</uploadedFileContents>
<text>Product</text>
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</jmol>

====Molecular Orbitals of the Transition State====

A molecular orbital diagram of the transition state is shown in '''Figure X'''.

[[Image:Bc608 MO Diagram.png|800px|thumb|centre|'''Figure X:''' ''A molecular orbital diagram for the transition state of the cycloaddition reaction'']]

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Predicted MO from LCAO''' || '''Calculated MO'''
|-
| [[Image:Bc608_HOMO-2_Button.png|100px]] || [[Image:Bc608_HOMO_Minus_two.png|150px]]
|-
| [[Image:Bc608_HOMO-1_Button.png|100px]] || [[Image:Bc608_HOMO-1.png|150px]]
|-
| [[Image:Bc608_HOMO_button.png|100px]] || [[Image:Bc608_HOMO.png|150px]]
|-
| [[Image:Bc608_LUMO_button.png|100px]] || [[Image:Bc608_LUMO.png|150px]]
|}

The HOMO and LUMO of the transition state are both symmetric with respect to the plane. The HOMO is comprised of the HOMO of ethylene and the LUMO of cis-butadiene.

The Woodward-Hoffman rules state that a 6π electron cycloaddition reaction will occur suprafacially via a Hückel transition state under thermal conditions. The molecular orbitals of the transition state show that the orbital overlap occurs via the one face of both the butadiene and ethylene components (i.e. the reaction is suprafacial) and thus the reaction is allowed.

====Geometry of the Transition State====

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || Value
|-
| '''C-C-C''' angle in butadiene fragment || 122.0
|-
| '''C-C-C-C''' dihedral angle in butadiene fragment || 0.0
|-
| '''C-C''' bond lengths || 1.39 (ethylene fragment), 1.38 (terminal C-Cs of butadiene fragment) and 1.41 (central C-C of butadiene fragment)
|-
| '''C-H''' bond lengths|| 1.09
|-
| '''Bond forming/breaking distance''' || 2.26
|}'''Figure X''': ''A table to show the geometrical parameters between the optimised transition state. All angles are in ° and are accurate to ± 0.05. All bond lengths are in Å and are accurate to ±0.01''

Typical C-C bond lengths are shown in '''Figure X'''.

{| class="wikitable" border="1" style="text-align: centre;"
|-
| '''Parameter''' || '''Length /Å'''
|-
| Typical sp3 C-C bond length || 1.53 <ref name="Sp3CC">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orpen and R. Taylor, ''Journal of the Chemical Society, Perkin Transactions'' 1987, '''2''', S1-S19 {{DOI|10.1039/P298700000S1}}</ref>
|-
| Typical sp2 C-C bond length || 1.33<ref name="Sp2CC">Wade, L.G. in ''Organic Chemistry'', '''2006''', 6th edition, Pearson Prentice Hall, 279</ref>
|-
| Van der Waals Radius of Carbon || 1.70<ref name="VDW">A. Bondi, ''Journal of Physical Chemistry'' 1964, '''68''', 441-451 {{DOI|10.1021/j100785a001}}</ref>
|}'''Figure X''': ''A table to show literature values for some parameters''

The bond breaking/forming lengths are significantly shorter than twice the Van der Waals radius of carbon (3.40Å), therefore there must be an attractive interaction in between the carbons involved in bond forming and bond breaking. However, this bond length is still larger than the typical C-C σ bond length, therefore the bond has not yet fully formed.

The central C-C bond of butadiene is showing significant double bond character and is shorter than typical C-C σ bond lengths (1.41Å cf. 1.53Å) but longer than typical C-C double bond lengths (1.41 cf. 1.33Å). The length of the central C-C bond however is still longer than the terminal C-C bonds, suggesting it has less double bond character than the terminal C-C bonds. Therefore, the transition state is likely to be a relatively early transition state, because the transition state resembles the reactants more than the products.

====Activation Energy====

In order to calculate the activation energy for the Diels Alder reaction, the energies of the reactants were summed, so that the number of atoms between reactants and products did not change. For cis-butadiene, the most stable conformer was used (trans-butadiene).

The summed reactant energies totalled -234.580 Hartree.

The activation energy is therefore 95.1 KJ mol-1.

===Diels Alder reaction of Malelic anhydride and Cyclohexa-1,3-diene===

====Optimisation of transition states====

Malelic anhydride and cyclohexa-1,3-diene were optimised using semi-empirical AM1 calculations. These structures were then used to draw a guessed structure of the exo and endo transition states. Centres that were to become sp3 in the product were made sp3 like in the transition state by setting relevant angles to 109.5°. Groups that sterically clashed were bent away from each other. These guessed exo and endo transition state structures were then optimised to a Berny transition state at the AM1 level of theory. Once found, the transition states were reoptimised using a DFT-B3LYP/6-31G(d) calculation. The starting materials were then also reoptimised at the higher level of theory. '''Figure X''' shows the results of the transition state optimisations.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Structure'''|| '''Starting Geometry''' || '''Log file of the AM1 optimisation''' || '''Log file of the DFT-B3LYP calculation'''|| '''Structure of TS at the DFT-B3LYP/6-31G(d) level of theory'''|| '''Energy / Hartree'''
|-
| Endo Transition State||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Endo_TS2_starting_geom.mol</uploadedFileContents>
<text>Endo TS Guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8145}} || {{DOI|10042/to-8135}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDOTSFINALGEOM.mol</uploadedFileContents>
<text>Endo TS Final Structure</text>
</jmolAppletButton>
</jmol>
||-612.68339680
|-
| Exo Transition State ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_Exo_TS_startinggeom.mol</uploadedFileContents>
<text>Exo TS guessed Structure</text>
</jmolAppletButton>
</jmol>
|| {{DOI|10042/to-8146}}|| {{DOI|10042/to-8147}} ||
<jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_TS_FINAL_GEOM.mol</uploadedFileContents>
<text>Exo Final Structure</text>
</jmolAppletButton>
</jmol>
|| -612.67931095
|}

The endo form is 10.7 KJ mol-1 more stable than the corresponding exo form at the DFT-B3LYP/6-31G(d) level of theory. This may be initially somewhat surprising, considering the endo form is more strained than the exo form. The main contribution to this difference in strain is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_Exo_Strain.png|thumb|centre|300px|''The exo transition state'']]
||
[[Image:Bc608_Endo_Strain.png|thumb|centre|300px| ''The endo transition state'' ]]
|} '''Figure X:''' ''The source of increased strain in the endo transition state.''

It can be seen that the two fragments are brought closer together in the endo transition state than in the exo transition state and thus the endo transition state is more strained. Despite the endo transition state being sterically more strained, it is nevertheless more stable than the exo transition state. Therefore, there must be a dominating, stabilising electronic effect operating that stabilises the endo transition state more than the exo transition state.

====IRC analysis of transition states====

Now the exo and endo transition states have been found, an IRC calculation must be performed to see which minimas the transition states connect. An AM1 IRC calculation was performed on the AM1 optimised structures, with the Hessian recalculated at every IRC iteration. The IRC calculations were performed both forwards and backwards.

IRC analysis of the endo transition state {{DOI|10042/to-8189}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_ENDO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
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</jmol> to the <jmol>
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<uploadedFileContents>Bc608_ENDO_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Endo product</text>
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IRC analysis of the exo transition state {{DOI|10042/to-8190}} was found to connect the <jmol>
<jmolAppletButton>
<uploadedFileContents>Bc608_EXO_STARTING_MATERIALS_IRC.mol</uploadedFileContents>
<text>Starting Materials</text>
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</jmol> to the <jmol>
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<uploadedFileContents>Bc608_EXO_TS_PRODUCTS_IRC.mol</uploadedFileContents>
<text>Exo product</text>
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</jmol>

====Electronic stabilisation: Secondary orbital overlaps?====

In order to explain the stability of the endo transition state, often "secondary orbital overlaps" in the transition state are invoked. The proposed orbital overlaps in the transition state are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_ENDOTSCARTOON.png|thumb|centre|150px|''Proposed orbital interactions in the endo transition state'']]
||
[[Image:Bc608_EXOTSCARTOON.png|thumb|centre|150px| ''Proposed orbital interactions in the exo transition state'' ]]
|} '''Figure X:''' ''The "textbook" orbital overlaps that result in the stabilisation of the endo transition state. Red dotted lines denote secondary orbital interactions, whereas black dotted lines denote primary orbital interactions.''

The molecular orbitals of the two transition states were generated in order to attempt to visualise this secondary orbital overlap effect. The HOMOs of the transition states are shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608_HOMO_ENDO_TS.png|thumb|centre|300px|''The HOMO of the endo transition state'']]
||
|[[Image:Bc608_HOMO_EXO_TS.png|thumb|centre|300px| ''The HOMO of the exo transition state'' ]]
|} '''Figure X:''' ''The HOMOs of the exo and endo transition states''

Careful examination of the HOMOs shows that there is no contribution from the carbon based carbonyl pZ orbitals in the HOMOs of either transition state. Therefore, if the secondary orbital overlap effect does exist, it must exist in a different molecular orbital. Examination of the LUMO+2 of the endo transition state does show a secondary orbital overlap effect. This is shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
| [[Image:Bc608 ENDO LUMO Plus 2.png|thumb|centre|300px| ''The LUMO +2 of the endo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the endo transition state''

Overlap can be seen between the carbon based carbonyl pZ orbitals with the diene based carbon pZ orbitals. These secondary orbital overlaps are not present in the LUMO +2 of the exo transition state as shown in '''Figure X'''.

{| class="wikitable" border="0" style="text-align: centre;"
|-
|[[Image:Bc608_Exo_LUMO+2.png|thumb|centre|300px| ''The LUMO +2 of the exo transition state'' ]]
|} '''Figure X:''' ''The LUMO+2 of the exo transition state''

The secondary orbital overlap effect in the LUMO+2 appears to be stabilising the endo transition state significantly. This initially seems somewhat confusing. Molecular orbital theory tells us:

# The LUMO+2 is unfilled according to the Aufbau principle.
# Only the stabilisation of occupied orbitals confers electronic stability upon a molecule.

If statements 1 and 2 are correct, then stabilisation of the LUMO+2 should be irrelevant and there should be no electronic stabilisation of the endo transition state. However, this is not what is observed computationally, therefore one of the two statements must be incorrect, implying molecular orbital theory is a simplification of reality.

If we look again at the Aufbau principle, it implies that molecular orbitals are either filled, half filled or completely unfilled. In other words, the orbital occupancy is either 0, 1 or 2 electrons. This essentially gives a "black" or "white" picture of orbital occupancy. Things are very rarely "black" or "white" in reality and are more often than not one of the many shades of grey in between.

Molecular orbitals can be thought of as electronic degrees of freedom of the system that arise as solutions to Schrödinger's equation. An analogy can be drawn between molecular orbitals and normal modes of vibrations - they are both degrees of freedom of the system. Electrons are distributed amongst the degrees of electronic freedom, just as vibrational energy is distributed amongst the normal modes. The LUMO+2 is an electronic degree of freedom of the endo transition state. Therefore, there is a probability associated with the LUMO+2 degree of freedom being occupied - it may be large, but it is not necessarily zero. The LUMO+2 will therefore be partially occupied in both transition states. Orbital occupancy should therefore not be thought of digitally as either 0, 1 or 2.

This partial occupancy of the LUMO+2 '''must''' exist in both transition states for the endo TS to be electronically more stable than the exo TS. The LUMO+2 is more stable for the endo transition state because it puts electron density between the two fragments that are coming together.

====Conclusions====

There is a partial occupancy of molecular orbitals above the HOMO and orbital occupancy can not be thought of digitally. A secondary orbital overlap effect exists in the partially occupied LUMO+2 of the endo transition state, which stabilises the endo transition state electronically. The secondary orbital overlap effect does not exist in the LUMO+2 of the exo transition state. The difference in electronic stabilisation is greater than the difference in steric destabilisation. The endo transition state is more strained yet is overall more stable.

These calculations however have completely ignored the possibility of solvent stabilisation of the exo and endo transition states, as they have been performed in the gas phase.

{| class="wikitable" border="1" style="text-align: center;"
|-
| '''Transition State'''|| '''Dipole moment ± 0.005 / Debye'''
|-
| Endo || 6.72
|-
| Exo || 6.14
|}

The Hughes-Ingold rules <ref name="Ingold">C.K. Ingold in ''Structure and Mechanism in Organic Chemistry'', 1953, Comell University: Ithaca, NY</ref> states that an increase in solvent polarity will accelerate the rates of reactions where there is a build up of charge in the activated complex. The endo transition state has a higher dipole moment than the exo transition state and is therefore possibly more polar. One might therefore expect the endo product to be favoured by polar solvents, whereas the exo product would be favoured by non polar solvents. Further calculations would be required in order to test this hypothesis however.

==References==

<references/>