ChemWiki - User contributions [en]

Rep:Jyc08module3

2011-03-25T16:54:09Z

Jyc08: /* ''anti'' - 1,5-hexadiene optimisation */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry: [[Image:ANTI2 OPT.LOG|thumb|Description]]

Return to Main Page.

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry: [[Image:ANTI2 OPT 6 31G.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given: [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian. [[Image:CHAIR TS OPT MOD REDUNDANT.LOG|thumb|Description]]

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate. [[Image:CHAIR TS OPT MOD REDUNDANT UNFROZEN 2.LOG|thumb|Description]]

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown. [[Image:BOAT OPT FREQ FAIL.LOG|thumb|Description]]

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run. [[Image:BOAT OPT FREQ.LOG|thumb|Description]]

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transition state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculation to work.

Looking at the structures of the optimised chair and boat TS, it is difficult to predict which conformer of hexadiene will form as the product. Therefore the next section will involve an IRC calculation which will aim to optimise the TS to a minimum energy structure, and hence give the structure of the predicted product.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:53:07Z

Jyc08: /* ''Anti 2'' Ci Conformation */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry: [[Image:ANTI2 OPT.LOG|thumb|Description]]

Return to Main Page.

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry: [[Image:ANTI2 OPT 6 31G.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given: [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian. [[Image:CHAIR TS OPT MOD REDUNDANT.LOG|thumb|Description]]

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate. [[Image:CHAIR TS OPT MOD REDUNDANT UNFROZEN 2.LOG|thumb|Description]]

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown. [[Image:BOAT OPT FREQ FAIL.LOG|thumb|Description]]

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run. [[Image:BOAT OPT FREQ.LOG|thumb|Description]]

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transition state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculation to work.

Looking at the structures of the optimised chair and boat TS, it is difficult to predict which conformer of hexadiene will form as the product. Therefore the next section will involve an IRC calculation which will aim to optimise the TS to a minimum energy structure, and hence give the structure of the predicted product.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

File:ANTI2 OPT 6 31G.LOG

2011-03-25T16:52:33Z

Jyc08:

File:ANTI2 OPT.LOG

2011-03-25T16:52:00Z

Jyc08:

Rep:Jyc08module3

2011-03-25T16:51:16Z

Jyc08: /* Optimising the "Chair" Transition Structure */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given: [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian. [[Image:CHAIR TS OPT MOD REDUNDANT.LOG|thumb|Description]]

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate. [[Image:CHAIR TS OPT MOD REDUNDANT UNFROZEN 2.LOG|thumb|Description]]

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown. [[Image:BOAT OPT FREQ FAIL.LOG|thumb|Description]]

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run. [[Image:BOAT OPT FREQ.LOG|thumb|Description]]

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transition state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculation to work.

Looking at the structures of the optimised chair and boat TS, it is difficult to predict which conformer of hexadiene will form as the product. Therefore the next section will involve an IRC calculation which will aim to optimise the TS to a minimum energy structure, and hence give the structure of the predicted product.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:50:20Z

Jyc08: /* Mod Redundant Method */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian. [[Image:CHAIR TS OPT MOD REDUNDANT.LOG|thumb|Description]]

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate. [[Image:CHAIR TS OPT MOD REDUNDANT UNFROZEN 2.LOG|thumb|Description]]

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown. [[Image:BOAT OPT FREQ FAIL.LOG|thumb|Description]]

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run. [[Image:BOAT OPT FREQ.LOG|thumb|Description]]

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transition state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculation to work.

Looking at the structures of the optimised chair and boat TS, it is difficult to predict which conformer of hexadiene will form as the product. Therefore the next section will involve an IRC calculation which will aim to optimise the TS to a minimum energy structure, and hence give the structure of the predicted product.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

File:CHAIR TS OPT MOD REDUNDANT UNFROZEN 2.LOG

2011-03-25T16:49:55Z

Jyc08:

File:CHAIR TS OPT MOD REDUNDANT.LOG

2011-03-25T16:47:24Z

Jyc08:

Rep:Jyc08module3

2011-03-25T16:46:02Z

Jyc08: /* Optimising the "Boat" Transition Structure */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown. [[Image:BOAT OPT FREQ FAIL.LOG|thumb|Description]]

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run. [[Image:BOAT OPT FREQ.LOG|thumb|Description]]

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transition state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculation to work.

Looking at the structures of the optimised chair and boat TS, it is difficult to predict which conformer of hexadiene will form as the product. Therefore the next section will involve an IRC calculation which will aim to optimise the TS to a minimum energy structure, and hence give the structure of the predicted product.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:42:59Z

Jyc08: /* Optimising the "Boat" Transition Structure */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown. [[Image:BOAT OPT FREQ FAIL.LOG|thumb|Description]]

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run. [[Image:BOAT OPT FREQ.LOG|thumb|Description]]

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transition state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculation to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:41:21Z

Jyc08: /* Optimising the "Boat" Transition Structure */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown. [[Image:BOAT OPT FREQ FAIL.LOG|thumb|Description]]

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

File:BOAT OPT FREQ FAIL.LOG

2011-03-25T16:39:48Z

Jyc08:

Rep:Jyc08module3

2011-03-25T16:37:52Z

Jyc08: /* IRC (Intrinsic Reaction Coordinate) Method */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:28:34Z

Jyc08: /* Chair */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed: http://hdl.handle.net/10042/to-8192

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

The thermochemistry data will then be used to calculate the '''Activation Energy''' of the formation of both the chair and the boat transition state, from the anti 2 reactant.

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:24:54Z

Jyc08: /* Chair */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Vibrational frequencies of chair TS]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:23:17Z

Jyc08: /* cis-Butadiene and ethylene MOs */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE FREQ.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

File:BUTADIENE FREQ.LOG

2011-03-25T16:22:51Z

Jyc08:

Rep:Jyc08module3

2011-03-25T16:22:00Z

Jyc08: /* cis-Butadiene and ethylene MOs */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown. [[Image:BUTADIENE OPT.LOG|thumb|Description]]

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

File:BUTADIENE OPT.LOG

2011-03-25T16:21:18Z

Jyc08:

Rep:Jyc08module3

2011-03-25T16:19:50Z

Jyc08: /* Optimisation of Ethylene and cis-Butadiene TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure: [[Image:TS BICYCLIC OPTFREQ AM1.LOG|thumb|Description]]

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:17:45Z

Jyc08: /* Endo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]] [[Image:ENDO PDT OPT.LOG|thumb|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Special:Upload

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: http://hdl.handle.net/10042/to-8191

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:15:22Z

Jyc08: /* Endo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' by removing the inter-fragment bonds and setting the distance between the two fragments to 2.2 angstrom.

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: [[Image:ENDO PDT OPT.LOG|thumb|Description]]

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T16:13:15Z

Jyc08: /* Endo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened: [[Image:ENDO PDT OPT.LOG|thumb|Description]]

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

File:ENDO PDT OPT.LOG

2011-03-25T16:12:52Z

Jyc08:

Rep:Jyc08module3

2011-03-25T16:12:24Z

Jyc08: /* Exo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure: [[Image:EXO TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file: [[Image:EXO FREQ TS AFTER PDT OPT.LOG|thumb|Description]]

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

File:EXO TS AFTER PDT OPT.LOG

2011-03-25T16:11:21Z

Jyc08:

Rep:Jyc08module3

2011-03-25T16:10:39Z

Jyc08: /* Exo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary: [[Image:EXO PDT OPT.LOG|thumb|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

File:EXO PDT OPT.LOG

2011-03-25T16:09:24Z

Jyc08:

Rep:Jyc08module3

2011-03-25T16:07:42Z

Jyc08: /* Conclusion */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction and also to collect data to compare to experimental data.

A number of different methods were explored and it was found that different methods would work better with different structures.

These methods were then used to study the Diels Alder reaction, which allowed the calculation of the activation energies. The calculated values were seen to agree well with experimental data, which showed that these methods could quite accurately be used to predict such characteristics of reactions.

The calculations also allowed simulated IR spectra to be predicted, allowing comparison to experimental spectra; and visualised MOs enabled an explanation of observations such as the endo-selectivity of the Diels Alder reaction.

It might be interesting to further explore the relative energies of the maleic anhydride and cyclohexadiene transition state, after having found that many secondary orbital interactions are present in the unoccupied orbitals.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T15:57:18Z

Jyc08: /* Conclusion */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the powerful nature of computational methods to find the lowest energy structures of molecules and transition states, and using the information from these calculations, to be able to predict the preferred product of a certain reaction

different methods which can be employed to optimise molecules and structures of our choice. The very powerful methods which have been explored have demonstrated that observed experimental results such as activation energies of reactions, and stabilities of different conformers, can be accurately predicted.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T15:51:17Z

Jyc08: /* Conclusion */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

== Conclusion ==

This exercise has demonstrated the different methods which can be employed to optimise molecules and structures of our choice. The very powerful methods which have been explored have demonstrated that observed experimental results such as activation energies of reactions, and stabilities of different conformers, can be accurately predicted.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T15:50:57Z

Jyc08: /* Conclusion */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

==== Conclusion ====

This exercise has demonstrated the different methods which can be employed to optimise molecules and structures of our choice. The very powerful methods which have been explored have demonstrated that observed experimental results such as activation energies of reactions, and stabilities of different conformers, can be accurately predicted.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T15:44:24Z

Jyc08: /* Secondary Orbital Overlap in the Exo and Endo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

==== Conclusion ====

==References==

<references/>

Rep:Jyc08module3

2011-03-25T15:42:06Z

Jyc08: /* Secondary Orbital Overlap in the Exo and Endo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ '''Occupied MOs showing SOIs'''
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this inter-fragment electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS to be shorter''' than in the exo TS.

{| class="wikitable" border="1"
|+ '''Unoccupied Mos showing SOIs'''
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

The visualised '''unoccupied MOs''' showed a higher prevalence of SOIs than the occupied MOs, both in the exo and the endo TS structures. However, it can be seen that although the corresponding occupied MOs of the exo and endo TS have similar forms, the extent of interaction is always greater in the endo TS than the exo TS.

For example, '''MO 57 of the exo TS''' shows a single SOI between the two fragments which is anti-symmetric with respect to the place of symmetry. The corresponding '''MO 57 of the endo TS''' also shows this interaction, but the endo TS has an additional interaction. The additional interaction is "behind" the first interaction, also with anti-symmetric symmetry but with opposite phase of the first.

The other unoccupied MOs shown in the table also exhibit a larger extent of SOI in the endo than the exo form. Although these MOs are currently unoccupied, if electrons are added to the system, these bonding MOs would lead to a lowering in energy of the system and it is possible that the calculated energy difference between the endo and exo TS would be even greater than it already is.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T15:15:53Z

Jyc08: /* Secondary Orbital Overlap in the Exo and Endo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The '''MOs 36-60''' of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows selected MOs from those visualised:

{| class="wikitable" border="1"
|+ Occupied MOs showing SOIs
|-
| exo || endo
|-
| [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Endo 45.jpg|thumb|MO 45]]
|}

Out of all of the occupied MOs which were visualised from MO number 37 to 47, it was MO 45 that showed a clear difference between the '''endo''' and '''exo''' MOs. There is very clear '''presence of SOIs between the two fragments in the endo TS''', whereas there is a complete '''absence of this electron density in the exo TS'''.

The SOI seen in the endo TS is a bonding, in-phase interaction and therefore when it is occupied by electrons as it is in this case, it will lead to a lowering of the energy of the molecule. This can therefore explain why the endo TS is slightly lower in energy than the exo MO.

It can also be noted that this SOI in the endo TS '''increases the amount of electron density between the two fragments''' in the TS, and therefore it may be responsible for causing the '''inter-fragment distance in the endo TS''' to be longer than in the exo TS.

{| class="wikitable" border="1"
|+ Unoccupied Mos showing SOIs
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

It can be seen by comparing the '''exo TS''' and '''endo TS''' MOs shown in the table above that there are clear differences between the corresponding MOs in both TS.

==References==

<references/>

Rep:Jyc08module3

2011-03-25T14:46:05Z

Jyc08: /* Secondary Orbital Overlap in the Exo and Endo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The MOs 36-60 of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows the Mos which showed the presence of SOIs between the two fragments.

{| class="wikitable" border="1"
|+ MOs showing SOIs netween the MOs 36-60
! !! C1-C2 !! C2-C3 !!C1-C6 !!C3-C4!!C6-C5
|-
| exo || [[Image:Exo 45.jpg|thumb|MO 45]] || || || ||
|-
| endo || [[Image:Endo 45.jpg|thumb|MO 45]] || || || ||
|}

{| class="wikitable" border="1"
|+ MOs showing SOIs netween the MOs 36-60
! !! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

==References==

<references/>

Rep:Jyc08module3

2011-03-25T14:45:25Z

Jyc08: /* Secondary Orbital Overlap in the Exo and Endo TS */

The following exercise will involve the characterisation of '''transition structures''' on potential energy surfaces for the Cope rearrangement and Diels Alder cycloaddition.

The transition structures of molecules larger than triatomic systems will be studied. ''' Molecular mechanics''' and '''force field methods''' will not work well for the following structure determination because they cannot describe bonds being made and broken, or '''changes in the bonding type''' and '''electron redistribution'''.

'''Molecular orbital-based methods''' will be used instead which numerically solve the Schrodinger equation and locates transition structures based on the the local shape of the potential energy surface. The structure of '''transition structure''' will be found, '''reaction paths''' and '''barrier heights''' will be calculated.

== Cope Rearrangement of 1,5-hexadiene ==

The following exercise will involve the locating of the '''low-energy minima''' and '''transition structures''' on the C6H10 potential energy surface. This will determine the '''preferred reaction mechanism''' of the [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene.

The [3,3]-sigmatropic shift rearrangement of 1,5-hexadiene is known to occur with a '''concerted mechanism''' via either a '''chair or a boat''' transition structure. It was thought that the boat transition structure is higher in energy than the chair structure, and this has been proved by B3LYP/6-31G theory, both in term of activation energies and enthalpies.

[[Image:Chair vs boat.gif|Description]]

In the following exercise, these will be calculated using Gaussian.

=== Optimising the Reactants and Products ===

''The following exercise will involve the '''optimisation''' of a structure, the '''symmetry''' will be used to find its point group, the '''vibrational frequencies''' will be calculated and visualised and the '''potential energies''' will be calculated and compared to experimental values.''

==== ''anti'' - 1,5-hexadiene optimisation ====

The molecule 1,5-hexadiene was drawn in Gaussview 3 with , making sure that the central 4 C atoms all had anti-periplanar conformations. The structure was '''cleaned''' in Gaussview, to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>1 5 hexadiene anti.mol</uploadedFileContents>
<text>anti 1,5-hexadiene</text>
</jmolAppletButton>
</jmol>

This structure was then optimised Gaussian. The input file was set to do an '''optimisation''', with the method '''Hartree Fock''' and the basis set '''-331G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was opened and visualised in Gaussview 3. The '''Summary''' after the optimisation is shown.

[[Image:1 5 hexadiene anti opt summary.jpg|Description]]

The '''energy''' of this optimised structure can be seen to equal '''-231.68539619''' a.u., which is only accurate to 2 decimal places compared the energies given in Appendix 1.

The optimised structure was '''symmetrised''' and the '''point group C2h/C1''' was found:

[[Image:1 5 hexadiene anti opt symmetry.jpg|Description]]

It was decided that another attempt should be made in the optimisation of another anti conformation, to try and obtain an optimised molecule with a more accurate energy.

Another ''anti'' structure was drawn in Gaussview 3 with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 opt.mol2</uploadedFileContents>
<text>anti before optimisation </text>
</jmolAppletButton>
</jmol>

The method for optimisation was the same as before, '''Hartree Fock''' and the basis set '''3-21G'''. Under the '''LINK 0''' tab the %memb was set to be "'''%mem=250MB'''". This input file was submitted to SCAN.

When the calculation had finished, the checkpoint file was downloaded, and the following structure was obtained:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Hexadien anti 3 after opt.mol</uploadedFileContents>
<text>anti after optimisation </text>
</jmolAppletButton>
</jmol>

[[Image:Hexadien anti 3 opt.jpg|Description]]

The '''energy, -231.68907066''' a.u., of this optimised structure was now correct to 5 decimal places when compared to the ''anti 3'' structure in Appendix 1. The '''point group symmetry''' of C2h was also the same as that found in the Appendix.

==== ''Gauche'' - 1,5-hexadiene optimisation ====

A '''gauche''' form of 1,5-hexadiene was drawn in Gaussview, with the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche 4 before opt.mol</uploadedFileContents>
<text>gauche before optimisation</text>
</jmolAppletButton>
</jmol>

An input file was created using the '''same method''' as before, and submitted to SCAN.

When the calculation was completed, the structure was opened in Gaussview 3 and viewed:

[[Image:Gauche 4 opt summary symmetry.jpg|Description]]

The energy of '''-231.69166699 a.u.''' was found to agree with the energy of the ''gauche 2'' structure in Appendix 1 to 5 decimal places.

The '''symmetry''' found for this structure was '''C2''', compared to the Appendix structure which was C2 also symmetry.

'''Another gauche optimisation''' was done, using the same method and basis set.

The optimised structure gave the following energy and symmetry:

[[Image:Gauche5 opt summary symmetry.jpg|Description]]

The '''energy of -231.68961575 a.u.''' was the same as that reported in the Appendix for the ''gauche 5'' conformation, '''-231.68962''' a.u., to 5 decimal places.

The '''C1 symmetry''' was also the same as that found in the Appendix.

Considering both the ''anti'' and ''gauche'' conformations, it would be expected that the lowest energy conformation is the anti :

[[Image:anti angle.jpg|Description]] [[Image:Gauche angle.jpg|Description]]

The diagram shows the view down the central C-C bond, from which the dihedral angle is measured and thus the conformation of the molecule is determined.
It can be seen that in the case of the ''anti'' conformation, the two ends of the molecule are as far apart from each other (on the basis of the dihedral angle) as possible, whereas in the case of the ''gauche'' conformation, the two ends of the molecule are closer together, and therefore are expected to have a steric clash, causing unfavourable repulsive interactions.

However, considering the calculations that have been done in this exercise, it can be seen that the ''gauche'' conformations are actually lower in energy than the ''anti'' conformation:

{| class="wikitable" border="1"
|+ Calculated Energies of ''anti'' and ''gauche''
! Energy !! anti 3 !! gauche 2 !! gauche 5
|-
| Hartree || -231.6890707 || -231.691667 || -231.6896158
|-
| kcal/mol || -145,386.977 || -145, 388.6063 || -145, 387.3191
|}

The difference in energy between ''anti 3'' and ''gauche 2'' is 1.6292 kcal/mol where gauche is more stable than anti. This indicates that another effect is contributing to the relative energies of these two conformers.

The two things that must usually be considered to explain stability are ''sterics'' and ''electronics''. Since it has been proved from the data that sterics are not the predominant factor in the stabilisation of the gauche conformer with respect to the anti conformer, then the answer should lie in the electronics of this molecule.

It was found that literature<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, ''Mol. Phys.'', '''2002''', ''100'', pp. 441-446 </ref> had suggested that the discrepency between the expectation that ''anti''was more stable, and the result that ''gauche'' was actually found to be more stable, could be explained by considering interactions between the MOs.

It was cited that the observation of a more stable gauche conformer was due to interactions between the C=C pi-orbitals and the H atoms (or the C-H sigma*-orbitals).

==== ''Anti 2'' Ci Conformation ====

The structure of the anti 2 conformation of 1,5-hexadiene was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised structure had the following structure, energy and symmetry:

[[Image:Anti2 opt.jpg|Description]]

The '''energy of -231.69253506''' a.u. was found to agree with that given in Appendix 1, -231.69354 a.u. . The Ci symmetry was also found to be the same.

This optimised structure was then reoptimised at '''DFT B3LYP/6-31G''' level.

The optimised structure had the following energy and symmetry:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Anti2 opt 6 31G.mol</uploadedFileContents>
<text>anti 2 optimised using B3LYP/6-31G</text>
</jmolAppletButton>
</jmol>

[[Image:Anti2 opt 6 31G.jpg|Description]]

The energy of this optimised structure was slightly different to the energy given in the '''appendix, -234.611710''', therefore another attempt was made to optimise the ''anti 2'' conformer.

Once again, the 3-21G optimised structure was used to create an input file , with '''optimisation to a minimum''' as the job type, '''DFT/B3LYP''' as the method and '''6-31Gd''' as the basis set. This input file was submitted to SCAN.

The completed calculation gave the following structure:

[[Image:Anti2 opt 6 31Gd.jpg|Description]]

This energy is closer to the energy given in the '''appendix, -234.611710''', although there is still a slight difference.

Therefore it was decided to carry out a '''opt+freq''' calculation in order to attempt to minimise the energy slightly further, and also to obtain some thermochemistry information. The method was kept the same as '''DFT/B3LYP''' and the basis set '''6-31Gd'''.

The '''opt+freq''' calculation gave the following structure:

[[Image:Anti2 opt and freq 6 31Gd.jpg|Description]]

It can be seen that this optimisation did indeed further minimise the energy of the structure from the previous optimisation, although it is now slightly below the energy of that given in the appendix. '''No imaginary frequencies''' have been calculated, which indicates that this is a minimum energy structure of the ''anti 2'' conformer.

The energy of both ''anti 2'' structures optimised using the different methods of calculation cannot be directly compared because the total energies that are given following a calculation is highly dependent on the '''method''' that has been used for that calculation, and the '''details''' of the calculation such as the basis set. In this instance, two difference methods were used (Hartree-Fock and DFT), as well as different basis sets (3-21G and 6-31G).

However, it is possible to compare the two structures obtained from the different methods by comparing their '''geometries''' after the calculation.

[[Image:Anti2 geometry labelled.gif|Description]]

{| class="wikitable" border="1"
|+ Comparison of '''bond lengths''' of anti 2 from 3-21G optimisation, and 6-31G optimisation (angstrom)
! !! C1-C2 !! C2-C3 !! C3-C4 !! C4-C5 !! C5-C6 !! C1-H !! C2-H !! C3-H !! C4-H !! C5-H !! C6-H
|-
| 3-21G || 1.32 || 1.51 || 1.55 || 1.51 || 1.32 || 1.07 || 1.08 || 1.09 || 1.09 || 1.08 || 1.07
|-
| 6-31G || 1.33 || 1.50 || 1.55 || 1.50 || 1.33 || 1.09 || 1.09 || 1.10 || 1.10 || 1.09 || 1.09
|}

It can be seen that the higher level '''6-31G''' results in roughly the same C-C bonds lengths but slightly longer C-H bonds than the lower level '''3-21G'''.

The '''angles''' in both structures were almost indentical, including the angles around the double bond, and the dihedral angle about the central C-C-C-C .

Therefore, it can be concluded that the two different methods used to optimise ''anti 2'' both yield the same general structure, with very little change in the bond lengths and angles.

It can therefore be concluded that in this case and similarlysimple cases, it is reasonable to first carry out a rough optimisation using a lower level method, followed by a higher level method to give a better and more accurate optimised structure.

===== Frequency Analysis =====

As mentioned previously, the '''opt+freq''' calculation at the '''6-31Gd''' level yielded '''no imaginary frequencies''' which indicated that the structure was a minimum energy structure.

The '''logfile''' was then viewed to study the '''thermochemistry''' section of the file. http://hdl.handle.net/10042/to-8059

[[Image:Anti2 opt and freq 6 31Gd thermochemistry.jpg|Description]]

It is known that:

* '''(i)''' refers to the potential energy at '''0 K''' including the zero-point vibrational energy (E = Eelec + ZPE)
* '''(ii)''' refers to the energy at '''298.15 K''' and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans)
* '''(iii)''' contains an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions
* '''(iv)''' includes the entropic contribution to the free energy (G = H - TS)

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

'''(i)''' and '''(iii)''' are slightly different to the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) respectively.

The very first '''6-31G'''-optimised structure from the previous optimisations of ''anti 2'' to a minimum was also '''used to run a frequency calculation'''.

Frequency calculations give an indication of whether or not the preceding optimisation calculation was successful. When optimising to a ''minimum'', it is expected that all vibrational frequencies are positive. When optimising to a ''transition state'', it is expected that one of the vibrational frequecies is negative. If more than one of the vibrational frequencies are negative, it is an indication that the structure has not optimised fully. Negative vibrational frequencies are known as ''imaginary frequencies''.
The calculated values of vibrational frequencies are able to produce a simulated IR spectrum for the structure.

The method was set to '''DFT/B3LYP''' and the basis set '''6-31G''' was used. This input file was submitted to run in Gaussian.

When the job was finished, the log file was opened, and the '''summary''' file was viewed.

[[Image:Anti2 freq summary 6 31G.jpg|Description]] [[Image:Anti2 freq values 6 31G.jpg|thumb|Vibrational frequency values]] [[Image:Anti2 freq IR 6 31G.jpg|thumb|Simulated IR spectrum]]

It can be seen that no '''imaginary frequencies''' were found after the calculation. This indicated that the previous optimisation of the ''anti 2'' conformer had successfully optimised to a minimum energy structure.

The vibrational frequency values are given. The '''simulated IR spectrum''' is also given.

The '''output file''' was then opened to view the section titled '''"Thermochemistry"'''.

[[Image:ANTI2 FREQ 6 31G.LOG|thumb|Description]]

This section contained the following information:

[[Image:Anti2 freq thermochemistry 6 31G.jpg|Description]]

{| class="wikitable" border="1"
|+ Thermochemistry
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.416252 || -234.408951 || -234.408007 || -234.447898
|}

The values '''(i)''' and '''(ii)''' are now even further from the values given in the appendix, '''-234.469203''' and '''-234.461856''' (a.u.) than in the case shwon above.

Therefore it was decided that for the purposes of this exercise, and the exercises to follow, the '''thermochemistry of trans 2''' conformer will be taken to be those found previously, since they were the closest to the given values:

{| class="wikitable" border="1"
|+ Thermochemistry (a.u.)
! (i) Sum of electronic and zero-point energies!! (ii) sum of electronic and thermal energies !! (iii) sum of electronic and thermal enthalpies !! (iv) sum of electronic and thermal free energies
|-
| -234.469198 || -234.461863 || -234.460919 || -234.500718
|}

These values will be used in subsequent exercises to calculate the activation energies of the chair and boar TS structures.

=== Optimising the "Chair" Transition Structure ===

''This section will involve setting up a transition structure optimisation by 1.) computing the force constants at the beginning of the calculation 2.) using the redundant coordinate editor 3.) using QST2. The reaction coordinate will be visualised, the IRC (Intrinsic Reaction Coordinate) will be run and the activation energies will be calculated for the Cope rearrangement via the "chair" and "boat" transition structures.''

Firstly, an '''allyl fragment''' was drawn in Gaussview 3 and optimised using the method '''HF/3-21G'''.

The optimised '''structure and summary''' is given:

[[Image:Allyl opt summary.jpg|Description]]

Two of these allyl fragments were then used to draw a rough chair transition state, with the structure shown. The distance between the two fragments was set to roughly 2.2 angstrom.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts guess.mol</uploadedFileContents>
<text>rough chair transition state</text>
</jmolAppletButton>
</jmol>

This initial rough structure for the chair transition state was used to create an input file for the '''optimisation of the chair transition state'''. The calculation was set to '''"opt+freq"''', '''"optimisation to a TS (Berny)"''', '''"force constants once"''' and the additional keywords '''"opt=NoEigen"''' were added.

After this calculation had run, the log file was opened and the '''structure and summary''' was viewed:

[[Image:Chair ts opt and freq.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be equal to that given in the Appendix '''-231.619322''' a.u.

It can be seen that '''one imaginary vibration''' was calculated, as expected for a transition state.

The vibrational frequencies were viewed and it was seen that the imaginary frequency had a value of '''818 cm-1''', as shown. [[Image:Chair ts opt and freq vibration frequencies.jpg|thumb|Calculated vibrational frequencies of chair transition state]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT AND FREQ BERNY.LOG</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

==== Mod Redundant Method ====

The initial rough structure for the chair transition state was then used to carry out an '''optimisation of the transition state''' using the '''frozen coordinate method'''.

The '''"Redundant Coord Editor"''' was used, and coordinates were added by clicking on the icon. Two terminal carbon atoms from both fragments which are involved in one bond formation/breaking were selected, '''"bond"''' was chosen, and '''"freeze coordinate"''' was chosen.

The icon was used to generate another coordinate. This time the two terminal carbons on the other end of both fragments were chosen, and the same selections were made.

It was known that the distance between the terminal C atoms on the two fragments was '''2.2''' on one end, and '''2.31''' on the other end.

Now an input file was created for an '''"optimisation" to a "minimum"''' and it was seen that '''"opt=modredundant"''' was already in the input line. This was submitted to Gaussian.

After the calculation was done, the checkpoint file was opened and it was seen that the structure was very similar to the transition that was obtained in the earlier optimisation, when the calculation was for "opt+freq" and the optimisation was to a "TS(Berny)".

[[Image:Chair ts opt mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt mod redundant.mol</uploadedFileContents>
<text>transition state optimisation frozen</text>
</jmolAppletButton>
</jmol>

It could be seen that the '''energy''' was not yet the expected minimum energy of '''-231.619322''' a.u.

It was seen that the distances between the fragments, ie. the '''bond forming/breaking distances''' were still '''2.2 and 2.3''' angstrom.

[[Image:Chair ts opt mod redundant bond1.jpg|Description]] [[Image:Chair ts opt mod redundant bond2.jpg|Description]]

Now to '''unfreeze''' the reactive coordinate of the previously "frozen" optimisation, the "Redundant Editor" was used to create a '''new coordinate'''. Two of the C atoms which were frozen in the previous optimisation were now selected and '''"bond"''' and '''"derivative"''' were selected. The same was done for the other pair of C atoms.

Now a calculation was set up for '''"opt+freq"''' to a '''TS(Berny)''', and force constants was kept as "never". The name of the file was changed in Link 0 and this was submitted to Gaussian to calculate.

When this calculation had finished, the checkpoint file was opened, and the optimised transition state had the following structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair ts opt and freq mod redundant unfrozen.mol</uploadedFileContents>
<text>transition state optimisation unfrozen</text>
</jmolAppletButton>
</jmol>

The structure was summary was viewed:

[[Image:Chair ts opt freq summary.jpg|Description]]

The '''energy''' of this optimised chair transition state was seen to be the same as given in the Appendix '''-231.619322''' a.u.

It can be seen that one '''imaginary frequency''' had been found, and this was seen to have a value of '''-817.86''' cm-1, as shown.

[[Image:Chair ts opt freq vibration.jpg|thumb|Description]]

The imaginary vibration was visualised:

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR TS OPT FREQ MOD REDUNDANT UNFROZEN.LOG‎</uploadedFileContents>
<script>zoom 150; frame 23; vectors 4; vectors scale 2; color vectors purple; vibration 4</script>
<text>Imaginary vibration</text>
</jmolAppletButton>
</jmol>

In order to compare the final geometries of the structures after both methods of optimisation, the two structures were labelled as follows: (Berny on the left, Mod Redundant on the right)

[[Image:Chair ts opt and freq berny labelled.jpg|Description]] [[Image:Chair ts opt unfrozen labelled.jpg|Description]]

{| class="wikitable" border="1"
|+ Comparison of chair bond distances (angstrom) and angles (degree) between 2 opt. methods
! !! C11-C4 !! C12-C3 !! C11-C9 !! C9-C12 !! C4-C1 !! C1-C3 !! C-H !! C-H !! !! C3-C1-C4 !! C12-C9-C11
|-
| Berny || 2.02 || 2.02 || 1.39 || 1.39 || 1.38 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|-
| mod redundant || 2.02 || 2.02 || 1.40 || 1.40 || 1.39 || 1.39 || 1.07 || 1.08 || || 120.5 || 120.5
|}

It can be seen both methods of optimisation yield the same optimised structure as there is very little difference in the bond lengths and angles shown above.

Therefore it can be concluded that for a relatively small system such as the one under study here, our initial guess is close enough to the transition state structure for both the "optimisation to TS(Berny)" and "frozen cooradinates" methods to give a reasonable structure for the TS. The "frozen cooridnates" method is known to be a faster and cheaper method but in the current case, the system is relatively small and the TS structrue was relatively easy to predict and therefore both methods took roughly the same amount to calculate and gave very similar optimised structures.

=== Optimising the "Boat" Transition Structure ===

The method for optimisation will be the QST2 method, in which the reactants and products are specified and the calculation will interpolate between the two structures in order to find the transition structure between them. It is essential that the atoms in the reactants and products are numbered in the same way, and this will need to be done manually in this exercise, to ensure that the products molecule will correspond to the numbering that would be obtained if the reactant molecule had rearranged.

Firstly, the 6-31G optimised ''anti 2'' reactant molecule input file was opened. This molecule was copied and pasted into two new windows so that a reactant window and a product window could be viewed side-by-side.

The '''atoms labels were edited''' so that the labels for the product molecule corresponded to the labels for the reactant molecule.

[[Image:Boat opt freq fail numbering.jpg|Description]]

A calculation was then set up for '''"opt+freq"''' to optimise to the transition state '''"TS(QST2)"''', and the Link 0 was edited for 250MB. This was submitted to Gaussian to run.

When the calculation failed, the checkpoint file could not be opened, so the log file was opened instead. The failed structure is shown.

[[Image:Boat opt freq fail 5.jpg|failed structure]]

It is known that when the calculation linearly interpolated between the two structures, it simply translated the top allyl fragment and did not consider rotation around the central bonds. Therefore it is clear that the QST2 method will not be able to locate the boat transition structure, starting from the reactant and product structures shown above.

The original input file used for the above QST2 calculation was then used to '''modify the reactant and product geometries''' in order to get them closer to the boat transition structure.

The central '''C-C-C-C dihedral angle''' was modified to 0 degrees, and the two '''inner C-C-C angles''' were modified to 100 degrees.

It was found that the product molecule had to once again be '''re-numbered''' in order to correspond to the reactant molecule.

[[Image:Boat opt freq numbering.jpg|Description]]

This was set-up for another '''QST2''' calculation as before, and submitted to Gaussian to run.

When the calculation was successfully completed, the log file was opened in Gaussview 5 to give the following '''structure and summary''':

[[Image:Boat opt freq summary.jpg|Description]]

The '''energy''' agreed with that given in the '''Appendix, -231.602802''' a.u.

Only one imaginary frequency was found, as expected, and this was visualised.

[[Image:Boat opt freq vibration.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT OPT FREQ.LOG</uploadedFileContents> <script>zoom 150; frame 43; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary Vibration</text> </jmolAppletButton> </jmol>

It can be seen that although the QST2 method allows the transtion state to be found in essentially one step, more effort is required in preparing the relative structures of the reactant and product. Consideration must also be put into the form of the transition state and hence the form that the reactant and product molecules should take in order for the calculatin to work.

=== IRC (Intrinsic Reaction Coordinate) Method ===

The checkpoint file of the '''mod redundant optimised chair transition state''' was opened in Gaussview 5 and an input file was created. '''"IRC"''' was selected as a job type, '''"forward direction"''' and '''"once"''' were chosen and the number of points along the IRC was changed from 10 to '''50'''. This was submitted to Gaussian.

When the calculation was not able to complete but the log file was opened up to give the following '''structure and summary''':

[[Image:IRC inital summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>CHAIR IRC MOD REDUNDANT.mol</uploadedFileContents>
<text>IRC of chair</text>
</jmolAppletButton>
</jmol>

It was seen that this structure was optimised after 37 steps, however the '''final energy, -231.67232608''' a.u. did not match with any of the ''anti'' or gauche'' structures given in the appendix. It was noted that this energy was higher than any of the energies listed in the appendix, therefore it could be concluded that this calculation had not fully minimised the energy of the structure yet.

This could also be seen from looking at the '''IRC plot''' from the calculation, showing the change in the total energy during the course of the calculation, as well as the change in the gradient during the course of the calculation.

[[Image:Chair IRC mod redundant.jpg|thumb|IRC plot of initial IRC calculation]]

The first plot shows that the calculation is converging in the right direction because the total energy is decreasing.

The second plot shows that the gradient is also decreasing as expected, but the important thing to note is that it has not decreased to zero and therefore a stationary point has not been found following the calculation.

It was thus known that the structure obtained from this IRC calculation was '''not yet the minimum geometry''', so this structure was further '''minimised (i)'''. An input file was created to do an '''optimisation to a minimum''' as the job type, using the same method and basis set as previously, '''HF/3-21G'''.

After the minimisation, the following '''structure and summary''' was obtained:

[[Image:Chair i IRC mod redundant summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair i IRC mod redundant.mol</uploadedFileContents>
<text>(i) minimised IRC of chair</text>
</jmolAppletButton>
</jmol>

The '''energy''' of this structure was seen to match with the energy of the ''gauche 2'' conformer, '''-231.69167''' a.u. given in the appendix, therefore it could be concluded that this minimisation had successfully found the product conformer that the chair TS would form.

Another '''IRC calculation (iii)''' was run using '''100 points''', during which the force constants were set to calculate at '''every step''' of the iteration.

This method of IRC calculation is

[[Image:Chair iii 100 IRC mod redundant.jpg|Description]]
[[ Image:IRC iii 100 plot 1.jpg|thumb|(iii) Total energy during IRC calculation]] [[Image:IRC iii 100 plot 2.jpg|thumb|(iii) Gradient during IRC calculation]]

The energy of this structure agrees closely with the energy of the ''gauche 2'' conformer given in the '''appendix, -231.69167''', as expected

The IRC path shows that this calculation optimised the structure successfully.

The first plot of the '''total energy''' shows that the energy has been optimised to a minimum, and the second plot shows that the gradient has converged successfully to zero, to a stationary point.

If '''this IRC plot (iii)''' is compared to that obtained in the '''initial IRC calculation''', in which the force constants were only calculated once and only 50 steps were used, it can be seen that the total energy of (iii) reaches a lower value and has a more flat plateau. This probably owes to the fact that 100 steps were used in calculation (iii) whereas only 50 steps were used in the initial calculation.

The gradient from the initial IRC calculation does not reach zero, whereas the gradient in calculation (iii) reaches zero and also has a plateau. This is again owing to the larger number of steps used in (iii) and hence the calculation is allowed to continue until a stationary point has been found.

Therefore it can be concluded that it is preferable to use many steps in an IRC calculation because this gives the calculation the chance of reaching a stationary point. It was found in this system that 100 points were sufficient for the stationary point to be found, but this may be more for larger or more complicated systems.

=== Activation Energies ===

==== Chair ====

The checkpoint file of the '''mod redundant''' optimised Chair transition state was used to create an input file to re-optimise the structure at a higher level. The job type was chosen as optimise to '''"TS(Berny)"''', the method was '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"''' was added as additional keywords. This was submitted to SCAN.

The output file was viewed:

[[Image:Chair REOPT 6 31g SUMMARY.jpg|Description]]

It was seen that the energy of the optimised structure agreed with the energy given in the '''appendix, -234.556983 a.u.'''

The '''geometry''' of the '''3-21G''' optimised and the '''6-31G''' optimised structures were compared:

{| class="wikitable" border="1"
|+ Overall geometry of 3-21G vs. 6-31G optimised structures (angstrom)
! !! Distance between 2 fragments !! C-C bond lengths !! C-H bond lengths !! central C-C-C angle within fragment !! H-C-H angle within fragment
|-
| 3-21G || 2.02 || 1.40 || 1.08 || 120.5 || 113.8
|-
| 6-31G || 1.98 || 1.41 || 1.09 || 120.0 || 112.5
|}

It can be seen that the inter-fragment distance is shorter for the higher level '''6-31G''' than the lower level '''3-21G'''. The higher level '''6-31G''' gives slightly longer bond lengths, and slightly smaller angles. However, these differences in bond distances and angles between the two levels of theory are in fact very small, and it can be seen that the geometries of structures optimised using either method are indeed very similar.

Therefore it can be concluded that when optimising a particular structure it is reasonable to firstly optimise it at a lower level, which would give a good estimate of the optimised structure, and subsequently to optimise this structure using a higher level theory to give a more accurate optimised structure.

A '''frequency calculation''' was now going to be done in order to obtain the '''thermochemistry information'''.

The optimised structure was used to create the input file, setting the job type as '''"frequency"''', using the same method of '''"DFT/B3LYP/6-31G"''' and '''"opt=noeigen"'''.

The calculation gave the following structure:

[[Image:Chair freq after reopt 6 31Gd summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq after reopt 6 31Gd.mol</uploadedFileContents>
<text>Chair TS, after freq calc</text>
</jmolAppletButton>
</jmol>

The vibrational frequencies were viewed:

[[Image:Chair freq after reopt 6 31Gd vibrational value.jpg|thumb|Description]]

It was confirmed that the chair TS had been successfully found in the above optimisation due to the single negative vibrational frequency '''-565.54''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Chair freq AFTER OPT 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The log output file was viewed to look for the '''Thermochemistry''' information:
[[Image:Chair freq AFTER OPT 6 31Gd.out|thumb|Description]]

[[Image:Chair freq after reopt 6 31Gd thermochemistry.jpg|Description]]

The output file for '''Chair 3-21G''' is [[Image:CHAIR TS OPT AND FREQ BERNY.LOG|thumb|Description]]

The output file for '''anti2 3-21G''' is [[Image:Anti2 freq.out|thumb|Description]]

The output file for '''anti2 6-31G''' is [[Image:Anti2 freq 6 31Gd.out|thumb|Description]]

==== Boat ====

The structure obtained from the QST2 optimisation was used to create an input file for the re-optimisation of the boat transition state structure. The job type was chosen as '''opt+freq''' and the method was '''DFT/B3LYP 6-31Gd'''.

The reoptimisation and frequency calculation gave the following '''structure and summary''':

[[Image:Boat reopt and freq 6 31G summary.jpg|Description]]

The vibrational frequencies were viewed to check that this structure was indeed a transition state:

[[Image:Boat reopt and freq 6 31G vibrational frequencies.jpg|thumb|Imaginary frequency of boat TS]]

It can be seen that one negative vibrational frequency had been calculated, which corresponded to '''-530.58''' cm-1.

<jmol>
<jmolAppletButton>
<uploadedFileContents>BOAT REOPT FREQ 6 31GD.mol</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency</text> </jmolAppletButton> </jmol>

The logfile was then viewed to collect the '''thermochemistry''' data.

[[Image:BOAT REOPT FREQ 6 31GD.LOG|thumb|Description]]

[[Image:Boat reopt and freq 6 31G thermochemistry.jpg|Description]]

All of the '''Thermochemistry''' data was collected, from the ''anti 2, chair and boat'' optimisations at both the ''3-21G'' and ''6-31G'' levels.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466703
| align="center" | -231.461342
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402340
| align="center" | -234.396006
|-
| '''anti2'''
| align="center" | -231.692535
| align="center" | -231.539542
| align="center" | -231.532571
| align="center" | -234.611710
| align="center" | -234.469198
| align="center" | -234.396006
|}

From this data, the energies in Hartree were converted to kcal/mol and from this, the activation energies were able to be found.

{| border="1" cellspacing="1" cellpadding="10"
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| ''' '''
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.05
| align="center" | 33.17
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.95
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

This results were found to be in good agreement with those given in the Apendix. It can be seen that the higher level '''6-31G''' method gave results which were closer to experimental values than the lower level method, despite both methods giving very similar structures as discussed above.

This thermochemistry information confirms that the '''chair TS has a lower activation energy''' than the boat TS and thus, under kinetic conditions, the chair will be the predominant product.

== The Diels Alder Cycloaddition ==

The Diels Alder cycloaddition reaction between butadiene and ethylene will be the subject of the following exercise. This is a reaction in which the π-orbitals of the dienophile (ethylene) form new σ-bonds with the diene (butadiene), also via the π-orbitals of the diene.

The concerted and stereospecific mechanism of a Diels Alder reaction is determined by the nodal properties of the two reactant fragments. The reaction is '''allowed''' when the '''HOMO''' of one reactant can interact with the '''LUMO''' of the other reactant.

Interaction is dependant on the the amount of overlap of electron density, which in turn is determined by the MOs of each reactant. When the '''symmetry properties of the MOs''' of both reactant fragments are the same, then overlap is possible and the reaction is allowed.

Furthermore, substituted dienophiles might feature substituents with π-orbitals that might interact with the new double bond which is formed during the Diels Alder reaction, and if this interaction is able to stabilise a particular regiochemistry, then this regiochemistry may be the predominant product.

It is clear that the factors which control the nature of the transition state are quantum mechanical, therefore quntum mechanical methods will be used in the following exercise.

[[Image:Diels alder.gif|Description]]

It is known that during the reaction of cis-butadiene with ethylene, shown, the principle orbital interactions involve the '''π/π*-orbitals of ethylene and the HOMO/LUMO of butadiene'''. This is a '''[4s+2s]''' reaction because the butadiene has 4 electrons in its π-system and ethylene has 2 electrons in it π-system. The s refers to the symmetry of the orbitals of butadiene and ethylene with respect to the plane of symmetry going through both fragments.

In this case, it is the '''HOMO of ethylene and the LUMO of butadiene''' which interact to result in a reaction, and both of these are '''symmetric''',

The two new σ-orbitals which are formed in the product have '''antisymmetric''' symmetry.

=== cis-Butadiene and ethylene MOs ===

[[Image:Butadiene opt summary.jpg|thumb| Structure and summary of optimised butadiene reactant]] [[Image:Butadiene opt MO HOMO value.jpg|thumb|Energies of HOMO and LUMO of butadiene]] [[Image:Ethylene_opt_summary.jpg| thumb|Structure and summary of optimised ethylene reactant]]
[[Image:Ethylene opt MO HOMO value.jpg| thumb|Energies of HOMO and LUMO of ethylene ]]

Cis-Butadiene was drawn in Gaussview, and '''optimised to minimum''' using the '''semi-empirical AM1''' method.

The HOMO and LUMO of the optimised butadiene were visualised. The visualisation of these are shown in the table below.

The same optimisation was done for the '''ethylene''' reactant, to give the optimised structure and summary shown.

Below shows the '''HOMOs and LUMOs''' of both the butadiene and ethylene reactants.

{| align=centre
|+ ''' HOMO and LUMO of butadiene'''
| [[Image:Butadiene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Butadiene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Butadiene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Butadiene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

{| align=centre
|+ ''' HOMO and LUMO of ethylene'''
| [[Image:Ethylene opt MO HOMO front.jpg|thumb|HOMO front view]]
| [[Image:Ethylene opt MO HOMO top.jpg|thumb|HOMO top view]]
| [[Image:Ethylene opt MO LUMO front.jpg|thumb|LUMO front view]]
| [[Image:Ethylene opt MO LUMO top.jpg|thumb|LUMO top view]]
|}

It can be seen that, just as expected, the '''HOMO of the ethylene and LUMO of butadiene are both symmetric''' with respect to the mirror plane, whereas the LUMO of ethylene and HOMO of butadiene are anti-symmetric. Any interactions between the two reactant fragments must take place between these specific pairs in order to have the same respective symmetry and therefore orbital interaction.

=== Computation of the T.S. Geometry and Examination of the Reaction Path ===

The transition state of the Diels Alder cycloaddition has an envelope type structure in order to maximise the overlap between the π-orbitals of the ethylene and the π-orbitals of the butadiene.

[[Image:TS guess.gif|Description]]

The '''initial guess geometry''' can be obtained by building a bi-cyclic system and then removing a fragment of this bicyclic system.

The '''distance''' between the two reactant fragments must also be estimated initially.

This initial guess structure of the T.S. can then be optimised in order to characterise the transition structure, and also confirmed following optimisation.

Once the correct T.S. structure has been obtained, the '''HOMO and LUMO''' will be plotted.

==== Optimisation of Ethylene and cis-Butadiene TS ====

The bicyclic template in Guassview was used to draw the initial guess structure for the transition state. A CH2-CH2 frragment was removed, and a double bond was added to give the following structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Ts bicyclic optfreq HF 321G.mol</uploadedFileContents>
<text>Initial guess of TS structure</text>
</jmolAppletButton>
</jmol>

An input file was created using this structure, the job type was '''"opt+freq"''', the optimisation was set to '''"TS(Berry)"''' and the force constants were set to calculate '''"once"'''.

The method was '''Semi Empirical/AM1''' and the additional keywords '''"opt=noeigen"''' were added.
This optimised structure gave the following structure:

[[Image:Ts bicyclic optfreq AM1 summary.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1 vibrations.mol</uploadedFileContents>
<text>AM1 optimised TS</text>
</jmolAppletButton>
</jmol>

[[Image:Ts bicyclic optfreq AM1 vibration value.jpg|thumb| Vibrational frequencies using AM1 method]]
[[Image:Bicyclic geometry.gif|Description]]

The '''bond distances''' of the optimised structure are given below:

{| class="wikitable" border="1"
|+ Bond distances of TS (angstrom)
! Partly formed C-C bonds !! "sp3" C-C !! "sp2" C-C
|-
| 2.12 || 1.38 || 1.40
|}

The sp3 and sp2 bond distances have been assigned according to the hybridisation of the product structure.

It was found in literature<ref name="bondlengths">S. J. Stuart, M. T. Knippenberg, O. Kum and P. S. Krstic, ''Phys. Scr.'', 2006, '''T124''', 58 - 64 {{DOI|10.1088/0031-8949}}</ref> that an '''sp3''' C-C bond is 1.54 angstrom and an '''sp2''' C-C bond is 1.47 angstrom. The '''Van der Waals''' radius is 1.7 angstrom for a carbon atom.

Comparison of the literature values with the bond distances found from the optimisation show that the calculation gave shorter bond distances than expected for the sp3 and sp2 C-C bonds.

However, the '''partly formed C-C bond distance of 2.12 is much larger than either of the sp3 and sp2 C-C''' bonds distances from literature. This is expected, since these two bonds are only ''partially'' formed in the transition state, and they form during the ''approach'' of one reactant onto another and therefore these partial bonds are expected to be longer than normal bonds.

It can be seen that the '''partially formed bond distances are ''shorter'' than twice the VdW radius''', which indicates that there is ''some'' level of bonding present, even if the bonds are not yet fully formed.

The '''imaginary frequency''' of the optimised TS was found to be '''-955.62''' cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 83; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary vibration</text> </jmolAppletButton> </jmol>

It can be seen that the imaginary frequency, ie. the frequency of the transition state, involves the making and breaking of the two bonds on either side of the two fragments in a '''concerted motion''' and therefore it is '''synchronous'''.

The '''lowest positive frequency vibration''' was found to be '''asynchronous''' at 146.77 cm-1.
<jmol>
<jmolAppletButton>
<uploadedFileContents>TS BICYCLIC OPTFREQ AM1.LOG</uploadedFileContents> <script>zoom 150; frame 84; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Lowest positive vibration</text> </jmolAppletButton> </jmol>

The checkpoint file was used to visualise the '''HOMO and LUMO''':

[[Image:Diels alder MO values.jpg|Energy values of HOMO and LUMO]]

{| class="wikitable" border="1"
|+ HOMO and LUMO
| HOMO || [[Image:Ts bicyclic optfreq AM1 HOMO 1.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 2.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 3.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 4.jpg|thumb|a]] || [[Image:Ts bicyclic optfreq AM1 HOMO 5.jpg|thumb|a]]
|-
| LUMO || [[Image:Ts bicyclic optfreq AM1 LUMO 1.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 2.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 3.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 4.jpg|thumb|s]] || [[Image:Ts bicyclic optfreq AM1 LUMO 5.jpg|thumb|s]]
|}

It can be seen that the '''HOMO is anti-symmetric''' with respect to the plane of symmetry, whereas the '''LUMO is symmetric'''. On this basis, upon comparison with the symmetries of the HOMO and LUMO of both ethylene and butadiene above, it can be concluded that the '''anti-symmetric HOMO''' shown here is made from the interaction between the '''HOMO of butadiene and the LUMO of ethylene, both anti-symmetric'''; whereas the '''symmetric LUMO''' shown here is made from the '''LUMO of butadiene and HOMO of ethylene, both symmetric'''.

This can be confirmed by studying the visualised MOs, which validates this conclusion.

==== Regioselectivity of Diels Alder ====

The reaction between cyclohexadiene and maleic anhydride undergoes a Diels-Alder reaction from which two products are possible.

[[Image:Maleic and cyclo.gif|Description]]

The following exercise will involve the calculation of the transition state structures '''endo''' and '''exo''', and the subsequent study of their relative structures, geometries and finally their HOMO.

===== Exo TS =====

Firstly, the exo product was drawn in Gaussview 5 and optimised in order to generate the lowest energy structure. The job type was '''optimisation to a minimum''' and the method '''semi=empirical/AM1''' was used.

The optimised '''exo''' product had the following structure and summary:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo pdt opt.mol</uploadedFileContents>
<text>optimised exo product</text>
</jmolAppletButton>
</jmol>

[[Image:Exo pdt opt summary.jpg|Description]]

The optimised product molecule was then modified to make it resemble the exo transition state. The two sigma bonds which are formed during the reaction were removed from the optimised product structrue, and the distance between the two fragments were adjusted to roughly '''2.2 angstrom'''. The double bond in the optimised product were also changed to delocalised bonds.

The '''guess of the transition state''' structure:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Guess of exo transition state</text>
</jmolAppletButton>
</jmol>

This guess of the transition state structure was then optimised to a '''transition state''' using the '''B3LYP/6-31G''' method to give the following structure:

[[Image:Exo ts 6 31G after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol>

A '''frequency calculation''' was then run with this optimised structure, using the same method as the optimisation, to confirm that a '''transition state''' had indeed been found.

The frequency calculation gave the following '''summary''' file:

[[Image:Exo freq TS after pdt opt summary.jpg|Description]]

The '''calculated frequencies''' were viewed to confirm that a transition state had been calculated:

[[Image:Exo freq TS after pdt opt vibration value.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>EXO FREQ TS AFTER PDT OPT.LOG</uploadedFileContents> <script>zoom 150; frame 3; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Imaginary frequency of exo TS</text> </jmolAppletButton> </jmol>

The '''HOMO''' of the optimised ''exo transition state'' was visualised:

[[Image:Exo ts 6 31G after pdt opt MO values.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO angle.jpg|Description]][[Image:Exo ts 6 31G after pdt opt MO top.jpg|Description]]

<jmol> <jmolAppletButton> <uploadedFileContents>EXO GFPRINT 6-31G.LOG</uploadedFileContents> <script>frame 49; mo 1; mo cutoff 0.020; mo fill; mo nomesh; mo translucent</script> <text>HOMO</text> </jmolAppletButton> </jmol>

[[Image:Exo geometry.gif|Description]]

===== Endo TS =====

The ''endo product'' was drawn in Gaussian and optimised to give the lowest energy structure.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo pdt opt.mol</uploadedFileContents>
<text>Molecule 1</text>
</jmolAppletButton>
</jmol>

[[Image:Endo pdt opt summary.jpg|Description]]

This optimised structure was then modified to resemble the ''endo transition state'' ...

The guess structure is given:

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts opt freq 6 31Gd after pdt opt.mol‎</uploadedFileContents>
<text>guess of endo TS </text>
</jmolAppletButton>
</jmol>

This structure was used to create an input file with the job type '''Opt+freq''', the method '''DFT/B3LYP''' and the basis set '''6-31Gd'''. The additional keywords '''opt=noeigen''' were added, and this was submitted to SCAN.

When the calculation was complete, the file was opened:

[[Image:Endo ts opt freq 6 31Gd after pdt opt summary.jpg|Description]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>

It can be seen that the energy of this '''endo TS is lower in energy than the exo TS'''. The two different structures can be compared in terms of energy because the same method has been used in their calculations.

The fact that '''1 imaginary frequency''' was calculated was indicative that a true transition state had been reached.

The '''vibrational frequencies''' were visualised:

[[Image:Endo opt and freq TS after pdt opt vibrational frequencies.jpg|Description]]

<jmol>
<jmolAppletButton>
<uploadedFileContents>Endo.out</uploadedFileContents> <script>zoom 150; frame 93; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Visualise</text> </jmolAppletButton> </jmol>

The HOMO of the optimised endo structure was visualised:

[[Image:Endo MO values 1.jpg|Description]][[Image:Endo MO.jpg|Description]][[Image:Endo MO 2.jpg|Description]]

==== Comparison of Exo and Endo TS Structures ====

The energy of the '''exo''' TS was found in this exercise to be '''-612.49098''' a.u. whereas the '''endo''' was '''-612.68339''' a.u. which agreed with the expected result that the endo would be lower in energy. This is because it is known that the reaction of maleic anhydride and cyclohexadiene is a kinetically controlled reaction resulting in the endo product, therefore the endo TS should be lower in energy.

In order to compare the structural differences between the two transition states, the two structures were labelled as follows, and the '''bond distances''' and selected '''bond angles''' are given:

[[Image:Exo geometry.gif|right|Description]] [[Image:Endo geometry.gif|right|Description]]
{| class="wikitable" border="1"
|+ Bond distances optimised '''exo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.40 || 1.40 || 2.29 || 2.29 || 1.41 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.22 || 1.22 || 1.43 || 1.43
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''exo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 94.9 || 94.9 ||99.2 || 99.2
|}

{| class="wikitable" border="1"
|+ Bond distances optimised '''endo''' TS (angstrom)
! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5!!C4-C5!!C6-C7!!C3-C8!!C7-C8!!C4-C9!!C5-C10 !! C9=O !! C10=O !! C9-O !! C10-O
|-
| 1.40 || 1.39 || 1.39 || 2.27 || 2.27 || 1.39 || 1.52 || 1.52 || 1.56 || 1.48|| 1.48 || 1.20 || 1.20 || 1.40 || 1.40
|}

{| class="wikitable" border="1"
|+ Angles in optimised '''endo''' TS (degrees)
! C1-C6-C5 !! C2-C3-C4 !! C7-C6-C5 !! C8-C3-C4
|-
| 99.0 || 99.0 ||94.3 || 94.3
|}

Considering the bond distances of the two TS structures, it can be seen that the main difference lies in the distance between the two fragments, the '''C3-C4/C6-C5''' distance, and the bond distances involving the O atoms. The other bond distances are reasonably similar.

The inter-fragment distances are '''shorter in the endo''' TS than the exo. This can be explained by considering the possible '''secondary orbital interactions''' which are possible in the '''endo''' TS. This will be discussed in the next section.

Similarly, all of the bonds which involve O atoms are '''shorter in the endo''' TS than the exo. This can be seen by studying the MOs which are present on the C=O groups in both the exo and endo structures. It is clear that these MOs are '''larger''' in size on the '''endo''' than on the exo. Also, it can be seen from looking at the HOMOs depicted above that the central O within the ring shows some electron density in the case of the endo TS, whereas there is very little electron density around this O atom in the exo TS.
The fact that the '''endo structure shows more electron density''' in this area can explain the observation that the bonds lengths are shorter in this area for the endo TS. The increased amount of electron density is able to undergo interaction and in effect this increases the bond orders in question, making the bonds shorter.

It can be seen that the '''C1-C6-C5/C2-C3-C4''' bond angle is '''larger in the endo''' TS, whereas the '''C7-C6-C5/C8-C3-C4''' bond angle is '''larger in the exo''' TS. This can be rationalised by considering the position of the maleic anhydride fragment with respect to the two sides of the cyclohexadiene group (C1-C2 VS. C7-C8). It is obvious that when the maleic group is facing towards the C7-C8 side (exo), the C7-C6-C5/C8-C3-C4 angle will be larger (compared to the endo) due to repulsion; whereas when the maleic group is facing C1-C2, the C1-C6-C5/C2-C3-C4 angle will be larger.

==== Secondary Orbital Overlap in the Exo and Endo TS ====

It was seen in the previous section that there was a difference in the the inter-fragment distances between the exo and endo TS structures.

This can be explained using the idea of '''secondary orbital interactions'''<ref name="SOI">J. I. Garcia, J. A. Mayoral, L. Salvatella, ''Acc. Chem. Res.'', 2000, '''33''', 658 - 664 {{DOI|10.1021/ar0000152}}</ref>.
[[Image:SOI.jpg|right|Description]]

It is known that a process is ''allowed'' when the atomic orbital lobes are in-phase, and these are known as '''first-order orbital interactions'''.

Secondary orbital interactions occur when molecular orbitals, from groups of atoms which are not directly bonded, are able to interact. This usually affects the rate of a reaction, and the selectivity of the product during a reaction. As is the case here, it is the secondary orbital interations which helps explain why the endo product is lower energy and thus favoured over the exo product.

The picture<ref name="SOI"> </ref> illustrates a Diels Alder reaction between maleic anhydride and furan, which is very similar to the reaction under study in this exercise, if the furan is replaced with cyclohexadiene. The picture shows the two different ways in which the reactants can orientate themselves for reaction, giving the possibility of an endo or exo product.

It is suggested that the endo orientation allows for 4 attractive secondary orbital interactions (SOIs), whereas the exo allows for only 2 SOIs.

[[Image:SOI repulsive.jpg|right|Description]] <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo TS after pdt opt.mol</uploadedFileContents>
<text>Optimised exo transition state</text>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo ts reopt and freq 6 31Gd.out</uploadedFileContents> <script>zoom 150; frame 60; vectors 4; vectors scale 2; color vectors purple; vibration 4</script> <text>Optimised endo TS</text> </jmolAppletButton> </jmol>
It should be noted that repulsive SOIs are also possible, as shown<ref name="SOI"> </ref>, and these interactions also show that the endo TS is expected to be more stable than the exo TS.

In order to illustrate this, '''more MOs of the exo and endo''' TS were further studied, in addition to the HOMOs shown above. It was '''expected that the endo MOs would show a lot of''' interaction between the two fragments to indicate that secondary orbital overlap effects were taking place and therefore were leading to the previously mentioned effects on the bond distances and the bond angles, most notably the '''shorter inter-fragment distance in the endo''' TS.

The MOs 36-60 of both the optimised '''exo and endo''' TS were visualised to look for signs of secondary orbital interactions. The table below shows the Mos which showed the presence of SOIs between the two fragments.

{| class="wikitable" border="1"
|+ MOs showing SOIs netween the MOs 36-60
! !! C1-C2 !! C2-C3 !!C1-C6 !!C3-C4!!C6-C5
|-
| exo || [[Image:Exo 45.jpg|thumb|MO 45]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 45.jpg|thumb|MO 45]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

{| class="wikitable" border="1"
|+ MOs showing SOIs netween the MOs 36-60
! !! C1-C2 !! C2-C3!!C1-C6!!C3-C4!!C6-C5
|-
| exo || [[Image:Exo 59.jpg|thumb|MO 59]] || [[Image:Exo 58.jpg|thumb|MO 58]] || [[Image:Exo 57.jpg|thumb|MO 57]] || [[Image:Exo 54.jpg|thumb|MO 54]] || [[Image:Exo 51.jpg|thumb|MO 51]]
|-
| endo || [[Image:Endo 59.jpg|thumb|MO 59]] || [[Image:Endo 58.jpg|thumb|MO 58]] || [[Image:Endo 57.jpg|thumb|MO 57]] || [[Image:Endo 54.jpg|thumb|MO 54]] || [[Image:Endo 51.jpg|thumb|MO 51]]
|}

==References==

<references/>