ChemWiki - User contributions [en]

Rep:Mod:VilfyComp

2014-03-21T16:07:33Z

Dv1111: /* Final conclusions */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=anti-symmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bond lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both ends is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry selection rules). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second considered Diels Alder reaction was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculations were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs, if the correct orientation is attained. Secondary orbital interactions between the п-systems of the -CH=CH- and -(C=O)-O-(C=O)- fragments can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were added onto the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
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<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition an IRC calculation was carried out 'in both ways'. To ensure accurate results, 150 points were computed. The calculation terminated at 88 points i.e. almost half of the ones set to be computed, which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism from the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

The endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
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<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The exo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that this cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2.93 Å for the exo transition state and 2.87 Å for the endo state. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character (sp3 and sp2 respectively). All bonds except the one for the -HC=C- fragment were closer to the 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment due to partial double bond character. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained because the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better, are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous Diels Alder exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be regarded. It therefore is important to match orbitals up which have similar energies showing another reason why the HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

Despite some limitations, this study gave a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T16:05:00Z

Dv1111: /* Exo/Endo Summary */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
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<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
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</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=anti-symmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bond lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both ends is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry selection rules). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second considered Diels Alder reaction was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculations were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs, if the correct orientation is attained. Secondary orbital interactions between the п-systems of the -CH=CH- and -(C=O)-O-(C=O)- fragments can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were added onto the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition an IRC calculation was carried out 'in both ways'. To ensure accurate results, 150 points were computed. The calculation terminated at 88 points i.e. almost half of the ones set to be computed, which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism from the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

The endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The exo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that this cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2.93 Å for the exo transition state and 2.87 Å for the endo state. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character (sp3 and sp2 respectively). All bonds except the one for the -HC=C- fragment were closer to the 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment due to partial double bond character. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained because the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better, are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous Diels Alder exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be regarded. It therefore is important to match orbitals up which have similar energies showing another reason why the HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:58:20Z

Dv1111: /* Exo/Endo Summary */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=anti-symmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bond lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both ends is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry selection rules). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second considered Diels Alder reaction was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculations were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs, if the correct orientation is attained. Secondary orbital interactions between the п-systems of the -CH=CH- and -(C=O)-O-(C=O)- fragments can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were added onto the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition an IRC calculation was carried out 'in both ways'. To ensure accurate results, 150 points were computed. The calculation terminated at 88 points i.e. almost half of the ones set to be computed, which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism from the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

The endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The exo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that this cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2.93 Å for the exo transition state and 2.87 Å for the endo state. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:56:40Z

Dv1111: /* Endo Transition State */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=anti-symmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bond lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both ends is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry selection rules). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second considered Diels Alder reaction was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculations were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs, if the correct orientation is attained. Secondary orbital interactions between the п-systems of the -CH=CH- and -(C=O)-O-(C=O)- fragments can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were added onto the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
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<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition an IRC calculation was carried out 'in both ways'. To ensure accurate results, 150 points were computed. The calculation terminated at 88 points i.e. almost half of the ones set to be computed, which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism from the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

The endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
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<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:55:24Z

Dv1111: /* Transition State calculations for cyclohexa-1,3-diene and maleic anhydride */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=anti-symmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bond lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both ends is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry selection rules). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second considered Diels Alder reaction was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculations were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs, if the correct orientation is attained. Secondary orbital interactions between the п-systems of the -CH=CH- and -(C=O)-O-(C=O)- fragments can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were added onto the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
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<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition an IRC calculation was carried out 'in both ways'. To ensure accurate results, 150 points were computed. The calculation terminated at 88 points i.e. almost half of the ones set to be computed, which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism from the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:50:35Z

Dv1111: /* Transition state for the cycloaddition between cis-Butadiene and ethene */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=anti-symmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bond lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both ends is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry selection rules). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:46:47Z

Dv1111: /* Transition state for the cycloaddition between cis-Butadiene and ethene */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=anti-symmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both ends is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:43:47Z

Dv1111: /* Cis-Butadiene */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=anti-symmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:42:36Z

Dv1111: /* Cis-Butadiene */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis were calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:37:21Z

Dv1111: /* Cis-Butadiene */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (for the individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO dictate the reactivity and hence only these orbitals were considered for the purpose of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be accessed.
{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:35:03Z

Dv1111: /* Diels Alder pericyclic Cycloaddition */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part of this study.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:34:45Z

Dv1111: /* Diels Alder pericyclic Cycloaddition */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new single Carbon-Carbon bonds and one C=C double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat, can form both the exo and endo product. A successful Diels Alder reaction depends on the geometrical orbital overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of 4n+2 type and hence occur via Hückel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and analysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:30:34Z

Dv1111: /* Activation Energies */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtained by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:30:21Z

Dv1111: /* Activation Energies */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and the activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:28:37Z

Dv1111: /* Activation Energies */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and by giving a final summary or relevant energies to predict through which transition state the reaction will undergo. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:22:37Z

Dv1111: /* IRC of the boat transition state structure */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into, (100° and 0°), to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (for the chair), 45 (for the boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T15:11:02Z

Dv1111: /* Freeze Coordinate Method (chair) */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method was carried out in two steps and was different from the Hessian method in the sense that it does not calculate force constants. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:45:50Z

Dv1111: /* Hessian Method (chair) */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39 Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:45:31Z

Dv1111: /* Hessian Method (chair) */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for all ''TS (Berny)'' transition state calculations was ''opt=noeigen''. This keyword implied that only one negative (=imaginary) frequency was allowed in the computation. Frequency relates to the second derivative of the potential energy surface diagram and therefore relates to the curvature of the graph. The second derivative is negative at a local maximum of the graph which can be referred to as a transition state of the reaction mechanism i.e. exactly what these calculations are about.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:35:19Z

Dv1111: /* Chair and Boat Optimisations */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the ''QST2'' method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that was used for transition state calculations was ''opt=noeigen'', meaning that only one negative (=imaginary) frequency was allowed in the computation.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:31:58Z

Dv1111: /* 1,5-hexadiene anti conformer (Cianti2) */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. The obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play a key role.

For all calculations carried out so far it must be kept in mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:30:17Z

Dv1111: /* 1,5-hexadiene anti conformer (Cianti2) */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001'' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

<jmol>
<jmolApplet>
<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

<jmol>
<jmolApplet>
<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:26:35Z

Dv1111: /* Finding and rationalising the lowest energy conformer */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained are shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001''' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
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[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:18:40Z

Dv1111: /* The Optimisation of the Reactants and Products */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label) i.e (anti) or (gauche)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained is shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001''' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

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<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
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[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:16:52Z

Dv1111: /* Introduction */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of various transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained is shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001''' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

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<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
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</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:13:50Z

Dv1111: /* Introduction */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 positions i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of varioous transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained is shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001''' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
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[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

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<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
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[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:12:58Z

Dv1111: /* Introduction */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''chair'''or the '''boat''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 or 4 position i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of varioous transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained is shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001''' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T14:11:17Z

Dv1111: /* Computational Module 3 (Physical) */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data such as looking at molecular orbitals, was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''boat''' or the '''chair''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 or 4 position i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of varioous transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained is shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001''' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T06:32:47Z

Dv1111: /* Further Discussions */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''boat''' or the '''chair''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 or 4 position i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of varioous transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained is shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001''' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
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<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
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<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
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[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Final conclusions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>

Rep:Mod:VilfyComp

2014-03-21T06:31:14Z

Dv1111: /* Further Discussions */

=Computational Module 3 (Physical)=

This computational experiment introduces and analyses examples of two pericyclic reaction types - the Cope rearrangement, discovered by A. Cope <ref>Cope, A. C.; Hardy, E. M. ''J. Am. Chem. Soc.'' '''1940''', ''62 (2)'', pp 441-444</ref> and the Diels Alder cycloaddition discovered by O. Diels and K. Alder<ref>Diels, O.; Alder, K. ''Synthesen in der hydroaromatischen Reihe'' '''1928''', ''460'', pp 98-122</ref>. The two reactions were investigated by carrying out computational chemistry calculations, whereby generally speaking, respective structures were optimised, transition states were both calculated and analysed and other analysis and interpretation of the data was performed, as outlined in the following.

==The Cope Rearrangement Tutorial==

===Introduction===
The Cope rearrangement belongs to the category of pericyclic reactions, meaning that a cyclic transition state is present for molecules that undergo this rearrangement route. In the process of a [3,3]-sigmatropic shift an allyl group migrates. This migration occurs with a concerted bond formation and breakage at different sides of the molecule. The example below shows the intramolecular rearrangement of a 1,5 diene triggered by heat.

[[File:Cope rearrangement.PNG|centre]]

This Cope rearrangement involves the movement of 6 electrons (4n+2). The reaction is thermally allowed via Hückel Topology, bearing suprafacial components (as opposed to antarafacial which would have been the case for a photochemically triggered reaction). The rearrangement can undergo two different types of transition states, the '''boat''' or the '''chair''' (see below), whereby the '''chair''' is considered to be lower in energy due to less steric repulsions.

[[File:Chair, boat TSs.PNG|centre]]

The Cope rearrangement stands in equilibrium, however the addition of substituents to the 1,5-hexadiene at the 3 or 4 position i.e. where the bond breakage or formation would take place, changes this and favours certain rearrangement to others. A more substituted end product would be favoured in that case due to increased thermodynamic stability at the more substituted end.

The following exercises demonstrate how the different transition states of 1,5-hexadiene can be analysed by investigating respective activation energies of the rearrangements and energy differences of varioous transition states to predict reactivity. Furthermore, symmetry was considered to correctly confirm the optimised structures, and energy pathways were calculated to identify local energy minima.

===The Optimisation of the Reactants and Products===

This part looks at different conformers (anti and gauche) of 1,5-hexadiene and optimises their structures, finds transition states and characterises their symmetry by finding their corresponding point groups. The following calculations also investigate vibrational frequencies to identifiy and reconfirm whether an optimisation or transition state was calculated. As a general procedure for the following exercises, all structures were created on the '''Gausview''' software, thereafter cleaned and optimised using the '''Hessian''' method which can be either '''HF/3-21G''' i.e. Hartree-Fock or '''B3LYP/6-31G(d)''' i.e. '''DFT'''. The ''(label)'' refers to the label found in the script to prove and show which structure the calculated molecule corresponds to.

====1,5-hexadiene anti-periplanar conformer (anti1)====

1,5-hexadiene was created in '''Gaussview''' in its antiperiplanar form, meaning that the dihedral angle was set at 180º between the four central Carbon atoms. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Anti conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React anti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is a summary of the obtained data on the molecule.

{| class="wikitable" border="2"
|+
Table 1. 1,5-hexadiene (anti1) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69260235
|-
| Total Energy (kcal/mol) || -145,389.33408424
|-
| RMS Gradient Norm || 0.00001824 a.u.
|-
| Dipole Moment || 0.2021 Debye
|-
| Point Group || C2
|-
| Log File || [[FILE:2_ANTI.LOG]]
|}

====1,5-hexadiene gauche conformer (gauche4)====

1,5-hexadiene was created in Gaussview in its gauche form, meaning that this time the dihedral angle was set at 60º instead of 180º between the four central Carbon atoms. The optimisation energy was expected be higher than for the previous case due to larger steric interactions in the gauche form. The molecule was cleaned and optimised via the '''HF/3-21G''' method. The molecular structure can be found below.

<jmol>
<jmolApplet>
<title>Gauche conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 2. 1,5-hexadiene (gauche4) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69153032
|-
| Total Energy (kcal/mol) || -145,388.66137511
|-
| RMS Gradient Norm || 0.00002038 a.u.
|-
| Dipole Moment || 0.1281 Debye
|-
| Point Group || C2
|-
| Log File || [[File:4_GAUCHE.LOG]]
|}

As predicted, the anti-structure turned out to be lower in energy which was assumed to be, due to steric effects. The gauche conformation pushes the four central atoms of the six membered Carbon chain to a close structure, whereby steric repulsion can occur.

===Finding and rationalising the lowest energy conformer===

Gauche 3 is lowest in energy. The fact that it is a gauche structure shows that steric effects do not play the crucial role but stereoelectronics are even more relevant when about reducing the energy of the molecule to a minimum. This instance is due to the ''gauche effect''. The effect is particularly strong for the '''gauche 3''' conformer (refer to appendix 1 in script) as electron donation from the C=C bond (π-bond) can be very well transferred to the σ* C-H, due to the geometry the molecule is in i.e. 60º. This process is also known as sigma conjugation. A good overlap of the bonding and anti-bonding orbitals is achieved as shown in the figure below which gives rise to an increased stabilisation for this orientation of this molecular structure.

[[File:Gauche effect cartoon.PNG|centre]]

The gauche 3 conformer was therefore constructed on ‘‘Gaussview‘‘, cleaned and optimised via the '''HF/3-21G''' method. The molecular structure, as well as a summary of the energy values obtained is shown below.

====1,5-hexadiene (gauche) conformer (gauche 3)====
<jmol>
<jmolApplet>
<title>Gauche 3 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>React gauche3.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 3. 1,5-hexadiene (gauche3) optimisation - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69266122
|-
| Total Energy (kcal/mol) || -145389.37102573
|-
| RMS Gradient Norm || 0.00000927 a.u.
|-
| Dipole Moment || 0.3407 Debye
|-
| Point Group || C1
|-
| Log File || [[File:3_GAUCHE.LOG]]
|}

===1,5-hexadiene anti conformer (Cianti2)===
This exercise investigated the ''anti2'' conformer. First, the structure was drawn out in ''Gausview'', cleaned and optimised via the '''HF/3-21G''' method.

'''1,5-hexadiene Cianti2 conformer (HF/3-21G)'''

<jmol>
<jmolApplet>
<title>Anti 2 conformer</title><color>white</color><size>300</size>
<uploadedFileContents>C(i) anti2.mol</uploadedFileContents>
</jmolApplet>
</jmol>

Below is the summary of the significant data for this calculation:

{| class="wikitable" border="2"
|+
Table 4. 1,5-hexadiene (anti2) optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69253528
|-
| Total Energy (kcal/mol) || -145389.29199717
|-
| RMS Gradient Norm || 0.00001891 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2.LOG]]
|}

The Hessian method can be carried out in two different methods, the lower level one ('''HF/3-21G''') and the higher level one ('''B3LYP/6-31G(d)'''). The lower level method, as will be shown later on is very useful to get an idea of whether a calculation was set up correctly, as it usually runs faster. The Hartree-Fock method is also more useful to analyse the orbitals of the optimised product or the transition state. The higher level method (also known as '''DFT''') is more accurate when calculating energies, it runs longer, uses less to almost no assumptions in the energy calculation and usually gives better results, as opposed to the Hartree Fock method. The second step in the analysis of the 1,5-hexadiene Cianti2 conformer was about the reoptimisation of the output file of the '''HF/3-21G''' calculation in ''Gaussview'' by applying the '''DFT''' method.

'''1,5-hexadiene Cianti2 conformer (B3LYP/6-31G(d))'''

<jmol>
<jmolApplet>
<title>C(i) anti2 DFT calculation</title><color>white</color><size>300</size>
<uploadedFileContents>C(i)_anti2_DFT_recalculation.mol</uploadedFileContents>
</jmolApplet>
</jmol>

{| class="wikitable" border="2"
|+
Table 5. 1,5-hexadiene (anti2) optimisation ('''B3LYP/6-31G(d)''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(D)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.61170280
|-
| Total Energy (kcal/mol) || -145,338.56883933
|-
| RMS Gradient Norm || 0.00001326 a.u.
|-
| Dipole Moment || 0.0000 Debye
|-
| Point Group || Ci
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Due to a change in basis sets between the two methods, a comparison of respective energies cannot be made. The two methods are based on different assumptions with respect to the theory they refer to and thus only the energies of molecules that were optimised by the same method can be compared. Nevertheless, it can be seen that the difference in bond lengths, especially at the C=C double bond is different for the two Hessian methods. The difference may be minimal, however it cannot be disregarded. Furthermore, the bond angles differ slightly between the two optmisation methods. The point group stays the same, so does therefore the symmetry of the molecule for both optimisations.

The last part in the analysis of the 1,5-hexadiene Cianti2 conformer was to investigate the frequencies. The key point was to show that no negative i.e. imaginary frequency was obtained in the optimisation calculation, hence proving a successful minimisation as opposed to a transition state calculation, where negative frequencies are present. Frequency calculations were run via the '''DFT''' method at both room temperature([[File:C(I)_ANTI2_FREQUENCY_CALC.LOG]]) and at 0.001K([[File:C(I)_ANTI2_FREQUENCY_CALC_FOR_0K.LOG]]), giving the same IR spectra which is shown below.

[[File:IRrrrrrrrr.PNG|400px]]

The ''vibration data'' indicated 42 different vibrations that gave rise to the obtained IR spectrum. The highest frequency vibration of the in total 42 vibrations is shown below:

[[File:Highest frequency vibration.gif|450px]]

Furthermore, the thermochemistry data for these calculations was investigated and respective energies for the room temperature and 0K case were compared. The different energy contributions that were considered were ''sum of electronic and zero-point energies'' (=Epot at 0 K including the zero-point vibrational energy), ''sum of electronic and thermal energies'' (=E at 298K and 1 atm of pressure which includes contributions from the rotational, translational and vibrational energy modes at this temperature), ''sum of electronic and thermal enthalpies'' and ''sum of electronic and thermal free energies''.

{| class="wikitable" border="2"
|+
Table 6. Thermochemistry data for 1,5-hexadiene (anti2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K) !! Energy in a.u. at 0.001K
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.469212 || -147131.683317 || -234.469212
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.461856 ||-147127.067357 || -234.469212
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.460912 || -147126.474988 || -234.469212
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.500821 || -147151.518269 || -234469212
|-
|}

0K could not be achieved, as ''Gaussview'' did not accept it as keyword, therefore ''temperature=0.001''' was chosen. In the data of '''Table 6''' it can be seen that the energies for the measurements at 0.0001K were all the same, for all different considered sums of energies. the obtained value '''-234.469212''' is also equal to the energy obtained for the '''sum of electronic and zero-point energies''' which makes sense, since at 0K, the other energy sums do not play key role.

For all calculations carried out so far it must be kept mind that:
*No imaginary frequencies were observed for any of the calculations, meaning that optimisations were successfully calculated as opposed to transition states.
*The RMS gradient (shown in the summary tables) gave low values, indicating that minimum values were reached i.e. reconfirms a successful optimisation.
*All values obtained matched clearly with the values given in the Appendix in the script.
*The two Hessian methods '''DFT''' and '''Hartree-Fock''' cannot be compared in terms of their energy results.

===Chair and Boat Optimisations ===

The next exercise investigates 3 different computational methods applied on the chair and boat structure of cyclohexane. The three considered methods are the ''Hessian'' method, the ''freeze coordinate = redundant coordinate'' method which consist of two steps and the QST2 method.

====Hessian Method (chair)====

The exercise was started by the optimisation of an allyl fragment (C3H5). The product of this optimisation process and the corresponding data is shown below:

[[File:Allyl fragment snapshot.PNG|centre|frame|Allyl Fragment - C3H5)]]

{| class="wikitable" border="2"
|+
Table 7. Allyl Fragment Optimisation ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .chk
|-
| Calculation Type || FOPT
|-
| Calculation Method || UHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Doublet
|-
| Total Energy (a.u.) || -115.8230401
|-
| Total Energy (kcal/mol) || -72,680.07049406
|-
| RMS Gradient Norm || 0.00003049
|-
| Dipole Moment || 0.0292 Debye
|-
| Point Group || C1
|-
| C-C-C bond angle || 124º
|-
| Equal bond distances || 1.39Å
|-
| Log File || [[File:C(I) ANTI2 DFT RECALCULATION.LOG]]
|}

Thereafter a ''guess chair structure'' was constructed made out of two allyl fragments which had to be moved manually to take in the correct geometry and the transition state was calculated. On ''Gaussview'' this was achieved by selecting '''Opt+Freq''' as job type, whereby the structure was set to be optimised to a ''TS (Berny)'' and the Force constant was calculated once. The method was '''HF/3-21G''' and the keyword that is usually used for transition state calculations is ''opt=noeigen''.

An image of the guessed transition structure is given below, followed by an animation of the obtain transition state.

[[File:Chair guess bonds.PNG|frame|left| Chair transition state guess structure d1-2 = 2.3Å and d2-3 = 2.15Å]]

[[File:Cope_Re.gif|400px]]

The above animation on the right (if it is not activated please click on it) was obtained by investigating the vibrations obtained in the output (LOG) file of the chair transition state calculation. A negative, imaginary frequency was observed at -818.01cm-1. The two C-C bond lengths that were pointed out in the guess chair structure were minimsed to be 2.20Å ([[File:Chair_TransitionState_Method_1.LOG]]). The table below reconfirms the formation of the correct transition state, also due to its great accuracy with the values in ''Appendix 2'' in the script.

{| class="wikitable" border="2"
|+
Table 8. Chair Transition State ('''HF/3-21G''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932230
|-
| Total Energy (kcal/mol) || -145,389.7235481
|-
| RMS Gradient Norm || 0.00004545
|-
| Imaginary Frequency || -818.01 cm-1
|-
| Dipole Moment || 0.0011 Debye
|-
| Point Group || C2h (after symmetrisation of the molecule)
|}

Finally the '''Thermochemistry''' data for the chair structure was found and is shown below:

{| class="wikitable" border="2"
|+
Table 9. Thermochemistry (chair transition state - Hessian method)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.466696 || -145247.575679
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.461338 ||-145244.213483
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.460393 || -145243.62049
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.495201 || -145265.46284
|}

====Freeze Coordinate Method (chair)====

The second method that was investigated for the chair structure was the '''frozen coordinate''' method. This method is carried out in two steps. This method is different from the Hessian method in the sense that it does not calculate force constants. The frozen coordinate method rather differentiates along the reaction coordinate. This approach can turn out to be much quicker, especially when trying to optimise or calculate transition state structures, where the exact position of the atoms for the guess structure are far apart from the reality case. In this way, the unknown bond distances can be regarded as redundant at first, and once the structure has been correctly optimised, the distances can be calculated to obtain the final structure.

To carry out this method, the two allyl fragments were again matched up in the way to form a chair structure when combined. The first step was about selecting the distances between the terminal Carbon atoms as ''redundant coordinates'' and the structure was optimised. It was found that after this first optimisation, the bond distances between the terminal Carbon atoms were fixed at 2.2Å. In the second part another optimisation was run, whereby the previously fixed bond distances were relaxed to allow for the correct chair transition state to form. The keyword used in step 1 was '''Opt=ModRedundant'''. The final distances between the terminal Carbon atoms were again 2.02 Å. The transition state obtained was almost identical with the one obtained via the Hessian method. All results obtained are listed below.

{| class="wikitable" border="2"
|+
Table 10. Chair Transition State ('''Frozen Coordinate''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FTS
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.6932240
|-
| Total Energy (kcal/mol) || -145389.7241756
|-
| RMS Gradient Norm || 0.00003621
|-
| Imaginary Frequency || -818 cm-1
|-
| Dipole Moment || 0.0003 Debye
|-
| Terminal bond lengths || 2.02Å
|-
| Point Group || C2h (after symmetrisation of the molecule)
|-
| Log File || [[File:FREEZE_COORDINATE_OPTIMISATION.LOG]]
|}

The differences in the '''thermochemistry data''' between the two methods was negligible. The imaginary frequency at -818 cm-1 corresponded again to the Cope Rearrangement, whereby the vibration indicated the bon formation and breakage at the terminal Carbon atoms by undergoing a six atom transition state.

====QST2 Method (boat)====

The third investigated method was the '''QST2''' method which was applied onto the cyclohexane ''boat'' structure. The '''QST2''' method uses the labeling system of the molecules. It has to be computed in ''Gaussview'' which molecule transforms to which (here according to the mechanism of the Cope Rearrangement). When starting off with two 1,5-hexadiene molecules, the molecules have to be set to the correct angles and conformation to ensure the formation of a boat transition state. This was done by setting the dihedral angle at the 4x central C-C bonds to 0º and the angles between the terminal C-C-C bonds to 100º, as illustrated below.

[[File:Boat_0_100.PNG|centre|450px]]

Any other set angles and bond distances can lead to the formation of other transition states, such as the chair-like transition state shown below.

[[File:Qst2_not_boat.PNG|centre]]

As a result, a boat transition state was obtained with a bond distance of 2.14Å between the terminal Carbon atoms (2.78Å between the two central Carbon atoms). The following two tables show all relevant data on the boat transition state including energy and thermochemistry data.

{| class="wikitable" border="2"
|+
Table 11. Boat Transition State ('''QST2''') - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.60280200
|-
| Total Energy (kcal/mol) || -608073.20297156
|-
| RMS Gradient Norm || 0.00007080
|-
| Imaginary Frequency || -839.94 cm-1
|-
| Dipole Moment || 0.1579 Debye
|-
| Terminal bond lengths || 2.14Å
|-
| Point Group || C2v (after symmetrisation of the molecule)
|-
| Log File || [[File:QST2_BOAT.LOG]]
|}

'''QST2: Boat transition state (Cope Rearrangement - imaginary frequency)'''
(if the animation is not activated please click on it)
[[File:Boat_TS_imag.gif|450px]]

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (boat transition state - QST2)
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -231.450928 || -145237.681108
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -231.445299 || -145234.148856
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -231.444355 || -145233.556487
|-
| Sum of electronic and thermal free energies || G = H -TS || -231.479774 || -145255.78225
|}

====IRC of the boat transition state structure====

The IRC (Intrinsic reaction coordinate) is a tool to demonstrate the reaction mechanism a molecule undergoes starting with the starting material and ending with the formation of the product via the transition state. The IRC calculations were run by submitting an '''IRC''' job type (50 steps, calculate one way, calculate force constant always) into ''Gaussview'' with the H/F3-21G method for both the output (LOG-files) of the respective conformers. This computational tool was used in the case of both the chair and boat transition states, as can be seen in the diagrams below:

'''Calculations regarding the Chair structure'''
[[File:IRC chair movie.gif|450px|centre|frame|IRC calculation for the '''chair''' conformer]]

[[File:IRC chair diagrams.PNG|600px]]

To calculate the IRC for the chair conformer, only 44 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.61932230 a.u. and the minimum value for the RMS gradient was 0.00004545. The given point group for the product was observed to be C2. Therefore it can be concluded with respect to the formed structure and point group that the gauche 2 conformer is eqivalent to the product.

'''Calculations regarding the Boat structure'''

[[File:IRC boat better.gif|450px|centre|frame|IRC calculation for the '''boat''' conformer]]

[[File:IRC boat diagrams.PNG|600px]]

The IRC calculation was carried out on the ''LOG'' file which was obtained from the previous '''QST2''' calculation. To calculate the IRC for the boat conformer, only 45 steps of the indicated max. 50 steps were considered by the software. The corresponding minimum energy is -231.60280200 a.u. and the minimum value for the RMS gradient was 0.00007069. The given point group for the product was observed to be Cs. With regards to the given appendix in the script no structure can however be allocated with Cs symmetry and it becomes clear that '''Gaussview''' in the IRC job remodeled the structure that was submitted from the QST2 exercise, i.e. the form into which the ''Reactant'' and ''Product'' had to be modeled into to form the boat conformer instead of the chair. This could be avoided in future work, when carrying out the QST3 exercise instead or calculating the boat transition state via for example the Hessian method.

From the animations it can be clearly observed which one corresponds to the boat and which one corresponds to the chair, when looking at the final movement of the terminal Carbon atoms (in the boat structure both ends move to the same direction). The animations also show when the bond was formed i.e. locate the correct transition state. The diagrams for each conformer outline that a metastable product was formed due to the plateau region at the right end of the graph. This corresponds to a local energy minimum and the total energy of the product formed in this reaction. The RMS gradient plot shows the same trend and by approaching 0.000 asymptotically it can again be concluded that the correct product was formed. Nevertheless, there are three approaches that should be considered to absolutely confirm the formation of the final product.

#Running a further minimisation from the last step given (44 or 45) in the IRC calculation
#Increasing the number of steps for the IRC calculation when submitting into ''Gaussview''
#Calculating the Force constant at every step

The first step was carried out and it was observed that the same minimum energy was obtained as shows below table for the chair conformer.

{| class="wikitable" border="2"
|+
Table 13. Chair Transition State Reoptimisation of step 44 from IRC - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RHF
|-
| Basis Set || 3-21G
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -231.69166702
|-
| Total Energy (kcal/mol) || -145388.74715567
|-
| RMS Gradient Norm || 0.00000475
|-
| Dipole Moment || 0.3806 Debye
|-
| Log File || [[File:IRC_CALCULATION_PIC_44_OPT.LOG]]
|}

From the results above it can be seen that the reoptimisation does not make a large difference nor improvement. The second suggestion can be ignored as the maximum amount of steps was 50 and only 44 (chair), 45(boat) were needed for the IRC calculation, hence a larger amount of steps would not have made a difference. The third suggestion was not considered either, since the force constant was set to be ''always calculated'' and hence it would not have given a better result. Therefore only the first suggestion can give a better result, nevertheless no big energy difference results from it.

====Activation Energies====

The final exercise was about the consideration of the activation energies (=energy difference between the reactant and transition state structure) for both chair and boat conformers and a final summary of the results obtained in previous exercises. The first step was the reoptimisation of the lower level HF-3-21G) to a higher level i.e. '''B3LYP-6-31G(d)'''.

'''Chair reoptimisation'''

<jmol>
<jmolApplet>
<title>Chair_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>CHAIR higher level opt.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Chair reopti structure.PNG|right|frame|Chair-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 14. Chair Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.55698303
|-
| Total Energy (kcal/mol) || -147186.76048199
|-
| RMS Gradient Norm || 0.00001198
|-
| Imaginary Frequency || -565.54 cm-1
|-
| Terminal C atoms distance || 1.96755 Å
|-
| Dipole Moment || 0.0000 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (chair transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.414929 || -147097.620213
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.409009 || -147093.905356
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.408065 || -147093.312987
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

'''Boat Reoptimisation'''

<jmol>
<jmolApplet>
<title>Boat_higher level opt</title><color>white</color><size>300</size>
<uploadedFileContents>BOATreopti.mol</uploadedFileContents>
</jmolApplet>
</jmol>

[[File:Boat reopti structure.PNG|right|frame|Boat-like transition state (higher level method)]]

{| class="wikitable" border="2"
|+
Table 15. Boat Reoptimisation at higher level (B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -234.54309307
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00000193
|-
| Imaginary Frequency || -530.30 cm-1
|-
| Terminal C atoms distance || 2.20679 Å
|-
| Dipole Moment || 0.0613 Debye
|-
|}

{| class="wikitable" border="2"
|+
Table 12. Thermochemistry (Boat transition state - reoptimisation B3LYP-6-31G(d))
! Energy classification !! Respective equation !! Energy in a.u. (at 298K) !! Energy in kcal/mol (298K)
|-
| Sum of electronic and zero-point energies || E = Eelec + ZPE|| -234.402343 || -147089.722377
|-
| Sum of electronic and thermal energies || E(298K) = E(0K) + Evib + Erot + Etrans || -234.396008 || -147085.747104
|-
| Sum of electronic and thermal enthalpies || H = E(298K) RT || -234.395064 || 147085.154735
|-
| Sum of electronic and thermal free energies || G = H -TS || -234.443814 || -147115.745828
|}

The above calculations show in both cases (chair and boat) that more accurate results can be obtained with the higher level optimisation method. As already mentioned it is not possible however to directly calculate the energy values that are obtained from the two methods, as '''Hartree-Fock''' and '''DFT''' are based on different assumptions. All transition state optimisations were carried out on the '''anti 2''' conformer, hence its energy is considered as the ''energy of reactant'' for the activation energy calculations.

From the data obtained from the higher level reoptimisation the difference in energy can be calculated and teh activation energies can be worked out for both chair and boat structures. It was observed that the calculated energies differed between the lower level and higher level optimisation. The same was experienced in the case of the anti 2 conformer where different methods gave different energy values.

 

''' Summary of energy values (all units are given in a.u.) '''

{| 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" | -231.61932230
| align="center" | -231.466705
| align="center" | -231.461338
| align="center" | -234.55698303
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450928
| align="center" | -231.445300
| align="center" | -234.54309307
| align="center" | -234.402343
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.69253528
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.61170280
| align="center" | -234.469212
| align="center" | -234.461856
|}

 *1 hartree = 627.509 kcal/mol 

''' Summary of the activation energies to predict transition state selectivity (in kcal/mol) '''

{| 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" | 45.89
| align="center" | 44.87
| align="center" | 34.20
| align="center" | 37.07
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.82
| align="center" | 54.98
| align="center" | 42.13
| align="center" | 45.26
| align="center" | 44.7 ± 2.0
|}

 

Values labelled as '''Expt.''' refer to the ones given in the lab script. The results obtained are fairly within the range of where we would expect them from experiments with some deviations. This outlines once again the importance of computational chemistry and its precision that is extremely challenging to achieve in the lab. The values obtained in the calculations agree quite accurately with the ones quoted in the script. The second table highlights which transition state is preferred at respective conditions and it can be observed that the chair transition state is generally lower in energy and hence more stable. The values obtained from the higher level method '''((B3LYP 6-31G(d))''' method match the experimental values better than the ones obtaoned by the '''Hartree-Fock''' method. The reason is most probably because the higher level method uses less approximations, takes longer and thus provides results that are more precise.

==Diels Alder pericyclic Cycloaddition==

The Diels Alder reaction represents a classic type of periciclic rearrangement whereby 6π electrons are moved in a concerted fashion, forming two new Carbon-Carbon bonds and one double bond. This cycloaddition where a diene is attacked by a dienophile is triggered by heat and can form both the exo and endo product. A successful Diels Alder reaction depends on the symmetry overlap between the two molecules i.e. only if they are of the same symmetry (asymmetric + asymmetric '''or''' symmetric + symmetric) a reaction can occur. The cycloadditions considered in the following calculations are all of the 4n+2 type and hence occur via Huckel Topology in a suprafacial fashion. The following exercises consider two examples where the transition states of Diels Alder cycloaddition examples were modeled and annalysed by methods that were explored in the previous part.

===Cis-Butadiene and ethene cycloaddtion: Optimisation and Transition States===

Below is the reaction scheme for the investigated Diels Alder reaction of this exercise.

[[File:NEW Diels-Alder Scheme.gif|centre]]

====Cis-Butadiene====

[[File:Cis-butadiene.PNG]]

Cis-Butadiene and ethene were optimised and their orbitals (individual molecules and transition state) as well as their transition states including IRC analysis was calculated to understand the mechanism of the underlying reaction. Only the HOMO and LUMO molecular orbitals dictate the reactivity and hence only these orbitals were considered for the sake of this analysis.

Table 13 shows the HOMO and LUMO orbitals for cis-Butadiene ([[File:CIS BUTADIENE OPT+FREQ.LOG]]). To view the symmetry, the checkpoint file had to be opened.

{| class="wikitable" border="2"
|+
Table 13. HOMO and LUMO for cis-Butadiene
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Cic cyclobutadiene HOMO(mesh).PNG]]|| [[File:Cis butadiene LUMO orbital.PNG]]
|-
| Energy in a.u.|| -0.34382 || 0.01709 ||
|-
| Symmetry || Anti-symmetric || Symmetric
|}

Regarding the symmetry it can be concluded that the HOMO has one node (=asymmetric) and the LUMO has two nodes (=asymmetric). Cis-Butadiene was optimised by the '''AM1''' method and showed a '''C2v''' symmetry with an energy of '''0.04879734 a.u.'''.

====Ethene====

[[File:Ethene.PNG]]

Table 14 shows the HOMO and LUMO orbitals for ethene ([[File:ETHENE OPT+FREQ.LOG]])

{| class="wikitable" border="2"
|+
Table 14. HOMO and LUMO for Ethene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Ethene HOMO orbital.PNG]] || [[File:Ethene LUMO orbital.PNG]]
|-
| Energy in a.u. || -0.38777 || 0.05284
|-
| Symmetry || Symmetric || Anti-symmetric
|}

Both the cis-Butadiene and ethene optimisations were run via the '''AM1''' method which is important in order to be able to compare them and match the optimised structures together onto one window in ''Gaussview'' to be able to calculate a correct transition state. Ethene showed a '''D2h''' point group and an energy of '''0.02619024 a.u.'''. Having analysed the optimisations of both reactants in terms of their orbital symmetry , it can already been predicted that the reaction can occur since the HOMO of cis-Butadiene has the same symmetry as the LUMO of ethene and vice versa.

====Transition state for the cycloaddition between cis-Butadiene and ethene====

The transition state below was calculated by first drawing out the bicyclo[2,2,2]octane molecule and modifying it to the correct alignment to perform a transition state calculation ('''TS Berny''', '''DFT''' method). This was achieved by removing two -CH2 fragments and adjusting the remaining parts to obtain the two reactant molecules. The figure below shows the transition state between cis-Butadiene and ethene.The atoms labelled in blue are the ones that form the new six membered ring structure. The bond distances between 1-2 and 3-4 are 2.27 Å. This 'bond' distance may sound very large for two Carbon atoms with 154pm and 134pm being typical bon lengths for a C-C and C=C bond respectively. By knowing that the Van der Waals radius of a carbon atom is about 170pm, according to '''A. Bondi''' <ref>Bondi A. 'J. Phys. Chem.'' '''1964''', ''68 (3)'', pp 441-451</ref>, a distance of 227pm would be much less than 2x 170pm, hence an interaction between the two terminal Carbon atoms on both end is reasonable to confirm. Moreover, the distance between the two unlabeled Carbon atoms was measured to be 1.41 Å which reconfirms that a double bond will form at this point, as also predicted by the final product structure.

[[File:TS_TS_TS.PNG|frame|left|Transition state of cis-Butadiene and ethene]]

The animation below illustrates the transition state, further enhanced by the displacement vectors. The total energy of this system '''E(RB3LYP)''' was '''-234.54389657 a.u.'''. The observed imaginary frequency, indicative for a transition state was -525.48 cm-1.

[[File:Butadiene_ethene_TS_animation.gif]]

The lowest molecular vibration occurs at 135.70 cm-1, where an asynchronous vibration mode can be observed which contradicts the other vibrations that all lead to the conclusion of a concerted bond formation.

'''Asynchronous vibration at 135.70 cm-1'''

[[File:Lowest_frequency_animation.gif]]

In addition, three '''IRC''' calculations were run for this transition state, one '''forward only''' one '''reverse only''' and one for '''both ways''', as selected in the calculation settings under Job Type. The number of computed points was set to 150 for all three calculations. Effectively it was possible to separate the IRC path diagram into the two parts -forward and reverse as can be seen below.

{| class="wikitable" border="2"
|+
Table 15. IRC breakdown
! Direction !! Total energy along IRC plot !!
|-
| Both ways || [[File:IRC_both_ways.PNG|400px]]
|-
| Forward || [[File:IRC_forward.PNG|400px]]
|-
| Reverse || [[File:IRC_reverse.PNG|400px]]
|}

The key messages from these three diagrams are:
*The 150 points were not exceeded in any of the three calculations meaning that the minimum structure was obtained.
*The 'both ways' IRC path represents the sum of both ''forward'' and ''reverse'' spectra.
*In the 'both ways' RMS Gradient spectra a minimum is achieved at both ends of the reaction coordinate axis (x-axis) reconfirming that the final product has formed.
*The formation of the two bonds proceeds in a concerted manner.

Finally, the IRC animation for the ''both ways'' calculation is shown below which shows the full reaction between cis-butadiene and ethene starting with separate reactants and terminating with the obtained product. The table below shows the molecular orbitals of the HOMO and LUMO in the transition state:

[[File:IRC_bothWays_animation_close.gif]]

{| class="wikitable" border="2"
|+
Table 16. HOMO and LUMO for the Butadiene + ethene transition state
! Molecular Orbital !! HOMO !! LUMO
|-
| Image || [[File:Butadiene ethene TS HOMO.PNG|400px]] || [[File:Butadiene ethene TS LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30084 || 0.14255
|-
| Symmetry || Symmetric || Anti-symmetric
|}

According to the arguments given earlier in this section it can be concluded from the symmetry of the orbitals in '''Table 15''' which molecular orbitals interact to give the HOMO and LUMO in the transition state. Since the HOMO is symmetric it must have been formed by the LUMO of cis-Butadiene and the HOMO of ethene to account for correct symmetry (symmetry allowed). Therefore the LUMO in the transition state was formed by the HOMO of cis-Butadiene and the LUMO of ethene. The above shown molecular orbitals were obtained from calculations that used the '''Hartree Fock''' method, as opposed to the pure energy and transition state calculations which were calculated via the '''DFT''' method. It is therefore plausible to conclude that different symmetry of the orbitals could have been expected when using different methods with different assumptions.

===Transition State calculations for cyclohexa-1,3-diene and maleic anhydride===

The second Diels Alder reaction that was considered, was the cycloaddition of cyclohexa-1,3-diene and maleic anhydride. As before the two starting materials were optimised first, using the '''DFT- B3LYP 6-31G(d)''' method this time. The two optimised structures were then pout together into a single window on ''Gaussview'' and aligned in the correct angle and distance to allow for both the exo and endo product (two possibilities) to form. The transition states were then analysed also by looking at the IRC method. Optimisation and energy calculation were carried out via the '''DFT- B3LYP 6-31G(d)''' method and molecular orbitals were extracted from separate '''HF 3-21G''' calculations. Maleic anhydride (below) was constructed on ''Gaussview'' and the molecular orbitals were obtained from the .chk file after correct optimisation.

====Maleic anhydride====

[[File:Maleic_Anhydride.PNG]]

{| class="wikitable" border="2"
|+
Table 17. HOMO and LUMO for Maleic Anhydride
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:Maleic anhydride HOMO.PNG|400px]] || [[File:Maleic anhydride LUMO.PNG|400px]]
|-
| Energy || 0.44751 || 0.02546
|-
| Symmetry || Symmetric || Anti-symmetric
|}

The total energy of maleic anhydride was observed to be '''-379.28954412 a.u.''' (according to the '''DFT''' method) The HOMO was identified as symmetric and the LUMO as anti-symmetric with respect to the plane of the board. As before, this information about the orbital symmetry is highly relevant as it dictates the manner in which the Diels Alder cycloaddition will proceed and can predict reactivity, hence very powerful.

====Cyclohexa-1,3-diene====

[[File:CYCLOOOOO.PNG]]

The total energy of cyclohexa-1,3-diene was calculated via the '''DFT''' method and gave '''-233.41893623 a.u.'''. The HOMO and LUMO for cyclohexa-1,3-diene are shown in the table below. The HOMO was identified to be anti-symmetric and the LUMO symmetric. This meant that the chosen starting materials could be matched up perfectly, since the HOMO of maleic anhydride and the LUMO are both symmetric and vice versa, thus matching orbitals of the same symmetry together according to the symmetry allowed transitions, thereby forming bonds.

{| class="wikitable" border="2"
|+
Table 18. HOMO and LUMO for Cyclohexa-1,3-diene
! Molecular Orbital !!HOMO !! LUMO
|-
| Image || [[File:1 3 cyclohexadiene HOMO.PNG|400px]] || [[File:1 3 cyclohexadiene LUMO.PNG|400px]]
|-
| Energy in a.u. || -0.30292 || 0.13618
|-
| Symmetry || Anti-symmetric || Symmetric
|}

==== Two possible transition States (Maleic Anhydride + Cyclohexa-1,3-diene)====

The scheme below indicates the two possible product outcomes of the Diels Alder cycloaddition that were studied in this exercise.

[[File:New_Maleic.PNG|centre|frame|Diels Alder Reaction Scheme showing the formation of two possible products - Endo and Exo]]

The '''endo''' product is favoured over the '''exo''' product, due to an increased and better orbital overlap between the p-orbitals which only occurs if the correct orientation is attained. Secondary orbital interactions between the π systems of the -CH=CH- and -(C=O)-O-(C=O)- fragment can be observed for the endo product which count as stabilising interactions. The two optimised starting materials were the same window on ''Gaussview'' and the two transition states were generated via the Hessian method '''(DFT B3LYP/6-31G)''' after correctly aligning the reactants to one another. The same applied to the endo structure where the method was the same but the angle and distances were different. Data regarding the Input file is given in '''Table 19'''.

{| class="wikitable" border="2"
|+
Table 19. Exo and Endo input files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:Input_file_exo.PNG|300px]] || [[File:Input_file_endo.PNG|300px]]
|-
| Bond distance1-2. || 2.14Å || 2.12Å
|-
| Angle1-2-3 || 132.973º || 124.617º
|}

====Exo Transition State====

The exo transition state was calculated by modeling the two reactants into the correct symmetry and bond distance, as shown in '''Table 19'''. An 'opt+freq', 'TS(Berny)' calculation was carried out via the higher level Hessian method. The key numerical data for the exo transition state is given below.

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<title>Exo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Exo_JmolJ.mol</uploadedFileContents>
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</jmol>

{| class="wikitable" border="2"
|+
Table 20. Exo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.67931091
|-
| Total Energy (kcal/mol) || -384462.15423759
|-
| RMS Gradient Norm || 0.00001825
|-
| Imaginary Frequency || -448.80 cm-1
|-
| Dipole Moment || 5.5505 Debye
|}

In addition to the exo transition an IRC calculation was carried out 'in both ways' with 150 computed points to ensure accurate results. The calculation terminated at 88 points i.e. almost half of the one set to be computed which is an indication for the success of the computation. The IRC path diagram attained a minimum value at high reaction coordinates which again shows that the calculation was finished. The IRC path diagram can be observed below, alongside the animation, showing the reaction mechanism form the starting materials to the final exo product, via the above mentioned transition state.

[[File:IRC_path_exo.PNG|300px]]

[[File:EXOEXO.gif]]

====Endo Transition State====

the endo transition state was calculated via the same method as the exo one, and all relevant numerical data is given in '''Table 21'''. When creating the starting materials in ''Gaussview'' and setting them up to give the endo transition state, maleic anhydride was effectively twisted by 180º, while the diene was kept in the same position. The correct endo transition state was attained as proves the structure and data below.

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<title>Endo final product</title><color>white</color><size>300</size>
<uploadedFileContents>Endo final prodyct.mol</uploadedFileContents>
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{| class="wikitable" border="2"
|+
Table 21. Endo Product at higher level (TS (Berny) B3LYP-6-31G(d)) - summary table
! Category !! Result
|-
| File Type || .log
|-
| Calculation Type || FREQ
|-
| Calculation Method || RB3LYP
|-
| Basis Set || 6-31G(d)
|-
| Charge || 0
|-
| Spin || Singlet
|-
| Total Energy (a.u.) || -612.68339674
|-
| Total Energy (kcal/mol) || -147178.04439864
|-
| RMS Gradient Norm || 0.00001187
|-
| Imaginary Frequency || -447.11 cm-1
|-
| Dipole Moment || 6.1143 Debye
|-
|}

An IRC calculation was also carried out for the endo transition state which ended after 84 points of the initially 150 points set to be computed. The data shows a similar trend as in the exo case i.e. an IRC path plot that reaches a low minimum (plateau region) at high reaction coordinates, meaning that the calculation was successful. The IRC energy plot and animation can be watched below.

[[File:IRC_path_endo.PNG|300px]]

[[File:EndoIRCMovie.gif]]

====Exo/Endo Summary====

The Diels Alder cycloaddition can theoretically lead to the formation of two different products exo or endo. The xo product is thermodynamically favoured and the endo product kinetically. The fact that the endo product is almost always preferred to the exo means that thsi cycloaddition is dictated by kinetics. The key reason is the secondary orbital effect, whereby the C=Cπ orbitals can overlap with the C=Oπ* orbitals. This can only occur for the endo product as firstly the bond distance and secondly and most importantly the angles in the exo product are unfavourable to form any sort of stabilising orbital effects, as the maleic anhydride component points away from the π-electron rich C=C bond. The distance between -(C=O)-O-(C=O)- fragment and the HC=CH fragment of the diene was measured to be 2. 93 Å for the exo transition state and 2.87 Å for the endo state.. All bonds were in the range of 134pm to 154pm, meaning their bond length were in between single and double bond character. All bonds except the one for the -HC=C- fragment were closer to 154pm indicating that they will remain as single bonds. The opposite trend was observed for the -HC=CH- fragment. A better insight into the relevant bond lengths for both transition state and products is provided in the table below. From this data we can learn that the exo product is actually more strained, as the maleic anhydride group is located in close proximity to the H2C-C-H2 group, whereby in the endo state, the close proximity between the maleic anhydride group and the C=C double bond is preferred due to the stabilising orbital effect. The reaction is under kinetic control and the endo product is favoured.

{| class="wikitable" border="2"
|+
Table 22. Exo and Endo output files in ''Gaussview''
! Product !! EXO !! ENDO
|-
| Sketch || [[File:ExoTS123.PNG|300px]] || [[File:EndoTS123.PNG|300px]]
|-
| Bond distance1-2. || 2.29Å || 2.27Å
|-
| Angle1-2-3 || 94.786º || 98.991º
|-
| Key bond distances (products) || [[File:Exo_final_features.PNG|300px]] || [[File:Endo_final_features.PNG|300px]]
|}

Another factor that has to be considered to understand the bonding and reactivity in this reaction better are orbital overlaps. All HOMOs and LUMOs are of anti-symmetrical geometry which is different to the example before. Maleic anhydride is strongly electron withdrawing and therefore its HOMO - LUMO gap will be more destabilised. This is one reason to account for the larger differences in relative energies when comparing the relative energies given in '''Table 23''' to the ones data of the previous exercise. It can be assumed that the orbital overlap that gives rise to the bonding in the exo and endo product is due to the interaction of the HOMO of the diene and the LUMO of the dienophile (maleic anhydride) because of their strong agreement in shape. When assessing how strong and effective an orbital overlap is, relative orbital symmetry and energy have to be assessed. It is therefore key to match orbitals up that have similar energies which gives rise to another reason why HOMO of the diene should interact with the LUMO of the dienophile.

{| class="wikitable" border="2"
|+
Table 23. Exo and Endo molecular orbitals
! MO type !! EXO (HOMO) !! EXO (LUMO) !! ENDO (HOMO) !! ENDO (LUMO)
|-
| Image || [[File: HOMO_exo_(-0.24217).PNG|200px]] || [[File:LUMO_exo(-0.07839).PNG|200px]] || [[File:HOMO_endo_(-0.24230).PNG|200px]] || [[File:LUMO_endo_(-0.06773).PNG|200px]]
|-
| Energy in a.u. || -0.24217 || -0.07839 || -0.24230 || -0.06773
|-
| Symmetry || Anti-symmetric || Anti-symmetric || Anti-symmetric || Anti-symmetric
|-
|}

===Further Discussions===

*This study investigated the two types of periciyclic reactions via a large amount of examples to illustrated and highlight certain effects. In the Cope rearrangement it was found that the preferred transition state was the chair. In the concerted Diels Alder cycloaddition it was found that nearly all products formed were of endo geometry.

*It was vital to grasp that different methods such as Hartree-Fock and DFT use different assumptions and basis sets and therefore their energies cannot be directly compared. It was found that DFT gave more precise results, regarding energy calculations, since the calculations took longer as less assumptions were used and Hartree-Fock computations turned out to be more useful for the correct generation of molecular orbitals.

*A limitation of this study was that solvent effects were not accounted for in any of the transition state calculations. It can be assumed that more polar solvents would have been better in stabilising the transition state and may have lead to fast calculations, however these assumptions would need to be looked into and further investigated and quantified.

*A final remark would be that the calculations that were carried out in this study did not account for kinetic effects, such as an increase in temperature or pressure and the result of faster moving particles which could have lead to fast reaction rates and possibly various different reaction pathways and slightly differing transition states, as opposed to only a single one.

This study gave however a great insight and encouraged a better and in more depth understanding of pericyclic reactions as a whole and showed what limiting factors had to be considered in the study of more complex computational calculations, while using different methods and levels of theories.

==References==
<references/>