ChemWiki - User contributions [en]

Rep:Mod:mariuspontmercy

2010-12-16T01:21:30Z

Ly308:

'''''Note to Marker''': As most of the calculations are rather fast, they are mostly run on the laptop instead of on the college SCAN platform. As a result, most of these do not have a D-space link to the public repository. The major output files (eg .log) are hence uploaded as an "image" link and you can access it by clicking it/downloading it -- as per instructions given to us in Module 2. Some files are however too huge to upload, and some intermediate steps of the calculation are not uploaded as there are too many files and the sizes are large.''

=Cope Rearrangement=

Cope rearrangement is a percicyclic reaction of [3,3] sigmatropic nature, defined by its characteristic concerted migration of a group from one point of attachment to a conjugated system to another point of attachment. It involves the breaking and formation of one σ-bond each during the process. Here, the reactant, 1,5-hexadiene, is optimised to its minimum-energy (or most stable) conformation before a transition state is calculated. Intrinsic reaction coordinate (IRC) is finally used to relate the starting material/final product comformation to the transitions tate.

==Optimisation of Reactant==
===Conformers of 1,5-hexadiene: HF/3-21G===

1,5-Hexadiene was optimised with an “anti” linkage (i.e. antiperiplanar dihedral angle at its central four carbons), by using Hartree Fork (HF) and a basis set of 3-21G. The ''Anti-1'' conformer was obtained and symmetrised in Gaussview. Next, a gauche conformer (''Gauche3'') was obtained in a similar manner by altering its central dihedral angle. The results are summarized in table 1.1.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.1: Optimisation of 1,5-Hexadiene Conformers''
|----
|
|'''''Anti-1'''''
|'''''Anti-2'''''
|'''''Gauche-3'''''
|----
|Optimised Energy /Hartree
| -231.6926
| -231.6925
| -231.6927
|----
|Optimised Energy /kJ mol<sup>-1</sup>
| -614541
| -614541
| -614542
|----
|Point Group Symmetry
|C<sub>2</sub>
|C<sub>i</sub>
|C<sub>1</sub>
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Hexadiene_anti1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_anti2.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|----
|}

Counter intuitively, the ''gauche3'' conformer is more stable than the ''anti1'', even though the steric crowding in ''gauche3'' could lead to repulsions and destabilisation. Literature results, eg reported by Rocque et al <ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> have also famously proven that gauche conformers are more stable and preferred than anti ones. This is explained by the stereoelectronic effect, which involves stabilisating interaction between the π-electrons of the C=C and vinyl protons, otherwise known as CH-π interaction, thus stabilising ''gauche3'' instead.<ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> This is confirmed by performing another optimisation on an anti-conformer – ''anti2''. And indeed, it is observed that ''anti2'' and ''anti1'' are comparable in their energies, and ''gauche3'' is lower in energy.

To view optimisation files for anti1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI1.LOG

To view optimisation files for anti2: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI.2LOG

To view optimisation files for gauche3: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_GAUCHE3_OPT.LOG

===Optimisation of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

''Anti2'' conformer is optimised by B3LYP method, with basis set of 6-31G*. A summary and comparison of both methods (HF/3-21G and DFT/B3LYP/6-31G*) are shown in table 1.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.2: Comparison of optimisation properties of anti2 conformer, by different methods''
|----
|'''Method'''
|'''HF/3-21G'''
|'''DFT/B3LYP/6-31G*'''
|----
|Optimised Energy/Hartree
| -231.6925
| -234.6117
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614541
| -622284
|----
|'''Geometry'''
|[[Image:Hexadiene_anti2.jpg|400px]]
|[[Image:Hexadiene_anti2.jpg|400px]]
|----
|Bond lengths C1-C2 & C12-C13/Å
|1.33
|1.32
|----
|Bond lengths C2-C6 & C7-C12/Å
|1.50
|1.51
|----
|Bond length C6-C7/Å
|1.55
|1.55
|----
|Bond angles C1-C2-C6 & C7-C12-C13/°
|125.3
|124.8
|----
|Bond angles C2-C6-C7 & C6-C7-C12/°
|112.7
|111.4
|----
|Dihedral C1-C2-C6-C7 & C6-C7-C12-C13/°
|118.5
|114.7
|----
|Dihedral C2-C6-C7-C12/°
|180.0
|180.0
|----
|}

To view DFT optimisation file: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI2_DFT_opt.LOG
To view DFT freq file: {{DOI|10042/to-6215}}

As can be seen, the final energy calculated by DFT is lower than that by HF, and their geometry, though very similar, differs slightly e.g. dihedral angles at the sides, vinyl bond angles, and alkene bond angles.

===Thermochemistry of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

Thermochemistry data is obtained by doing a frequency calculation on the optimised structure, at the same level of theory DFT/B3LYP/6-31G*, at temperatures 0K and 298.15K respectively. Data is summarised in table 1.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.3: Comparison of Thermochemistry Data of Anti2 At 0K & 298.15K''
|----
|
|'''T=0K'''
|'''T=298.15K'''
|----
|Sum of electronic and zero point energies/Hartree
| -234.4663
| -234.4692
|----
|Sum of electronic and thermal energies/Hartree
| -234.4607
| -234.4619
|----
|Sum of electronic and thermal enthalpies/Hartree
| -234.4597
| -234.4609
|----
|Sum of electronic and thermal free energies/Hartree
| -234.4955
| -234.5008
|----
|}

To view opt+freq file at 0K: {{DOI|10042/to-6216}}

It is observed that the energies are higher for the system at 0K than at 298.15K, indicating that the molecule absorbs more heat at higher temperatures and thus thermal contribution to the energies increases.

==''Chair'' Product==

===Optimisation of Chair Product===

Two approaches are used to optimise the chair transition state structure, both at the level of theory of HF/3-21G, at default temperature of 198.15K and 1.0atm pressure. One was done using optimisation to TS(Berny) directly, calculating the force constant only once. The other was done using the redundant coordinate, first to freeze the breaking/forming bond at 2.2Å and optimise to minimum, then finally removing the freeze and optimise the whole structure to a minimum transition state TS(Berny). Results are summaried in table 1.4:

{| class="wikitable" border="1"
|+ ''Table 1.4: Comparison of Optimisation of Chair- Transition State with Different Conditions''
|----
|
|'''TS (Berny)'''<br />{{DOI|10042/to--6217}}
|'''TS (Berny) with Redundant<br />Coordinate''' <br />{{DOI|10042/to--6218}}<br />{{DOI|10042/to--6219}}
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_ts_TSBerny.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_TSBerny_(mod_red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Optimised Energy/Hartree
| -231.6193
| -231.6193
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614347.0313
| -614347.0313
|----
|Bond forming length/Å
|2.02043
|2.02069
|----
|Bond breaking length/Å
|2.02051
|2.02074
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -818
| -818
|----
|Vibration Figure
|[[Image:Chair_ts_HF_freq818.jpg|150px]]
|[[Image:Chair_ts_HF_freq818(mod-red).jpg|150px]]
|----
|}

Both sets of results yielded very similar outcomes, suggesting that either method is reliable and sufficient. Looking at the imaginary frequency at -818cm<sup>-1</sup>, it can be seen that the bond breaking and forming motions occur in a concerted manner simultaneously and asynchronously, corresponding to the definition of pericyclic sigmatropic reaction.

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

Intrinsic reaction coordinate (IRC) calculation was run to track the energy of the transition state along its reaction pathway, reaching a minimum value. The number of points along IRC was set at 100, and a minimum was reached successfully. This calculation is repeated for both force constant calculation for once and for always. See table 1.5

{| class="wikitable" border="1"
|+''Table 1.5: Comparison of IRC results for Chair-TS ''
|----
|'''Force Constant Computation'''
|'''Energy/Hartree'''
|'''Energy along IRC Path'''
|'''Gradient'''
|'''Minimum Structure'''
|----
|'''Once''' <br /> {{DOI|10042/to-6220}}
| -231.6851
|[[Image:Chair_IRC_(ONCE)_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_min_str.jpg|200px]]
|----
|'''Always'''<br /> {{DOI|10042/to-6221}}
| -231.6917
|[[Image:Chair_IRC_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_minimum_structure.jpg|200px]]
|----
|}

For the “force constant=once” calculation, even though IRC points set to 100, only 21 steps were performed by the programme and the energy graph did not plateau to a minimum, and its corresponding RMS gradient definitely did not reach zero. Hence, it did not successfully reach a minimum geometry. However, for the “force constant=always” calculation, the gradient has fallen to zero, it shows that the total energy of the structure has indeed fallen to a plateau minimum successfully, due to force constants being calculated at each and every step. This successful calculation can then be related to the gauche3 conformer obtained above, which has a comparable energy of -231.6927.

==Optimisation of ''Boat'' Product==

The boat transition state is optimised using yet another approach – with QST2 at the same level of theory. Here, it was ensured that the numbering of the atoms in reactant and that in product match exactly; otherwise, the calculation fails. Summary of optimisation is shown in table 1.6:

{| class="wikitable" border="1"
|+''Table 1.6: Optimisation results of Boat-Transition State by QST2''
|----
|Energy /Hartree
| -231.6028
|----
|Energy/kJ mol<sup>-1</sup>
| -614303.2667
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -840
|----
|Bond forming/breaking length/Å
|2.14
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Boat.mol</uploadedFileContents></jmolApplet></jmol>
|----
|D-space link
|{{DOI|10042/to-6223}}
|----
|}

Looking at the imaginary frequency at -840cm<sup>-1</sup>, the vibration motion stretches horizontally in plane in concerted and asynchronous manner, indicating again that the breaking and formation of bond is concerted and simultaneous, confirming the definition of Cope rearrangement.

==Activation Energy==

Thermochemical energies of chair/boat transition states are compared with reactant, and each corresponding activation energies for chair and boat pathways are calculated and compared, across T=0K and T=298.15K, all done at the same level of theory at DFT/B3LYP/6-31G*. The summary of energies are shown in table 1.7:

{| class="wikitable" border="1"
|+''Table 7: Comparison of Activation Energies of Chair & Boat TS at 0K & 298.15K''
|----
|
|
| colspan="4" | '''T=0K'''
| colspan="3" | '''T=298.15K'''
|----
|
| '''Electronic Energy'''
|'''Sum of electronic & zero-point energies/Hartree'''
|'''Activation Energy E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|'''E<sub>A</sub>(expt, 0K)/kcal mol<sup>-1</sup>'''
|'''Sum of electronic & thermal free energies/Hartree'''
|'''E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|----
|'''Chair TS'''
| -231.6193
| -234.4119
|0.0544
|34.11
|33.5+-0.5
| -234.4082
|0.0537
|33.67
|----
|'''Boat TS'''
| -231.6028
| -234.3990
|0.0673
|42.23
|44.7 +- 2.0
| -234.3960
|0.0659
|41.32
|----
|'''Reactant'''
| -231.6925
| -234.4663
|NA
|NA
|NA
| -234.4619
|NA
|NA
|----
|}

E<sub>A</sub> for T=0K is done by taking the difference between the sum of electronic & zero-point energies of reactant anti2 1,5-hexadiene and the chair/boat transition state, followed by converting the units from Hartree to kcal/mol (multiply by 627.509). E(A) for T=298.15K is done by the difference between their sum of electronic & thermal energies instead. The activation energy is found to be always higher for the boat than chair conformation, which agrees with expectation that chair-cyclohexane is usually the more stable form, with minimal steric clash due to the staggered H’s. In boat TS, the eclipsed hydrogens will tend to result in steric or torsional strain, causing destabilisation of the structure.

At T=0K, computational values found here are in good agreement with literature experimental values, with the values being just very slightly out of range from experimental data. This could be due to many trivial causes, especially since computational methods generate slightly different results each time it is run, or it could be due to the degree of accuracy imposed by the level of theory used. However, generally, the trend that chair-TS is more stable than its boat counterpart is well followed by computational methods here.

To view Chair-TS opt+freq by DFT:{{DOI|10042/to-6224}}{{DOI|10042/to-6225}} <br />
To view Boat-TS opt+freq by DFT: {{DOI|10042/to-6222}}{{DOI|10042/to-6226}}

=Diels Alder Reaction=

The Diels-Alder reaction is a [4+2] cycloaddition, between a conjugated diene and an dienophile, involving the concerted formation of two (or more) σ-bonds between the termini of two or more conjugated π-systems. The reaction investigated here is between cis-butadiene and ethylene; and the reaction between 1,3-cyclohexadiene and maleic anhydride is studied for its regioselectivity property.

==Ex 1: HOMO & LUMO of Reactants==

The HOMO and LUMO for reactants (cis-butadiene and ethylene) are obtained upon their optimised structure, done at the level of theory of semi-empirical/AM1. Their properties are summarised in table 2.1:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.1: HOMO & LUMO of cis-Butadiene & Ethylene (Reactants)''
|----
|
|'''Cis-Butadiene'''
|'''Ethylene'''
|----
|'''HOMO'''
|[[Image:YL_Butadiene_HOMO.jpg|150px]]
|[[Image:YL_Ethene_HOMO.jpg|150px]]
|----
|Energy /Hartree
| -0.34381
| -0.38777
|----
|Symmetry
|anti-symmetric
|symmetric
|----
|'''LUMO'''
|[[Image:YL_Butadiene_LUMO.jpg|150px]]
|[[Image:YL_Ehene_LUMO.jpg|150px]]
|----
|Energy /Hartree
|0.01708
|0.05284
|----
|Symmetry
|symmetric
|anti-symmetric
|----
|}

To view cis-butadiene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_MO.LOG <br />
To view ethylene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_MO.LOG

==Ex 2: Transition State Geometries==

===Optimisation/Geometry===

The transition state of the reaction is optimised with TS(Berny) with specified redundant coordinate method – the breaking/forming bonds are frozen to 2.1Å first and optimised to a minimum, before optimising with bond derivative to TS(Berny). The level of theory used is semiempirical/AM1. The optimisation summary shown in table 2.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.2: Optimisation of transition state''
|----
|Structure
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>TS_(mod-red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
|0.11163
|----
|Energy/kJ mol<sup>-1</sup>
|296
|----
|Bond forming length/Å
|2.1189
|----
|Bond breaking length/Å
|2.1196
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -956
|----
|Vibration
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|}

To view optimisation for TS: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:TS_OPT2_MOD-RED.LOG

The optimised geometry is further investigated in table 2.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.3: Geometry of optimised transition state''
|----
|
|'''Transition State'''
|'''Butadiene'''
|'''Ethene'''
|----
|Structure
|[[Image:TS_with_labels.jpg|350px]]
|[[Image:Butadiene_with_labels.jpg|250px]]
|[[Image:Ethene_with_labels.jpg|250px]]
|----
|Bond length C6-C4/Å
|1.4
|1.45
|NA
|----
|Bond length C1-C4/Å
|1.38
|1.34
|NA
|----
|Bond length C6-C8/Å
|1.38
|1.34
|NA
|----
|Bond length C11-C12/Å
|1.38
|NA
|1.33
|----
|Bond angle (C4-C1-C12 & C6-C8-C11)/
|99.3
|
|NA
|----
|Bond forming (C6-C11)/Å
|2.12
|NA
|NA
|----
|Bond breaking (C1-C12)/Å
|2.12
|NA
|NA
|----
|}

From literature <ref name="0031-8949">F.H. Allen, Olga Kennard, D.G. Watson, ''J. Chem. Soc. Perkin Trans. II'', 1987, S1-S18 </ref><ref name="dewarcc">M. J. S. Dewar and H. N. Schmeising, ''Tetrahedron'', 1960, '''11''', {{DOI|10.1016/0040-4020(60)89012-6}} </ref>, a typical sp<sup>3</sup> C-C bond is 1.544Å long, while sp<sup>2</sup> C-C bond is 1.338Å long, and that the van der walls radius of carbon is 1.7Å. Judging from the data obtained, the partially formed bonds (2.12Å) are all longer than typical sp<sup>3</sup> C-C bond but shorter than twice the van der waals radius of carbon. It thus suggests that there is indeed bonding/overlap, and since a bond is still in the process of forming, it is an incomplete C-C bond, and hence, longer than usual.

===Molecular Orbitals===
The HOMO and LUMO orbitals of the transition states are shown in table 2.4:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.4: HOMO and LUMO of transition state''
|----
|
|'''HOMO'''
|'''LUMO'''
|----
|MO
|[[Image:YL_TS_HOMO.jpg|150px]]
|[[Image:YL_TS_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.32392
|0.02315
|----
|Symmetry (wrt plane)
|antisymmetric
|symmetric
|----
|}

Comparing to the HOMO and LUMO orbitals of the reactants shown earlier, it can be seen that the HOMO of transition state consists of the antisymmetric HOMO of cis-butadiene and the antisymmetric LUMO of ethane, forming a π(butadiene, diene)-π*(ethylene, dienophile) interaction, typical of cycloadditions. Since 6π electrons are involved (i.e. 4n+2 where n=1), under thermal conditions it would proceed via the Huckel route with all suprafacial components, and hence the bond forms on the same face of both the butadiene and ethylene π systems.

The HOMO(butadiene)-LUMO(ethylene) interaction is more favourable than the other way round due to the smaller energy gap:
*HOMO(butadiene)-LUMO(ethene) = 0.39665 Hartree (smaller, more favourable)
*HOMO(ethene)-LUMO(butadiene)= 0.40485 Hartree (larger, not favoured)

===Vibrational Frequency===
{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.5: Vibrations of Transition State''
|----
|'''Frequency/cm<sup>-1</sup>'''
|'''Description'''
|'''Mode of Vibration'''
|----
| -956
|Synchronous and concerted bond formation
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|147
|Rocking: ethene unit rocks about butadiene unit.
|[[Image:TS_vib_freq-147.jpg|150px]]
|----
|}

From the imaginary frequency of vibration, it can be seen that both bonds form in a concerted synchronous manner, where the ethene carbons move towards or away from butadiene unit together. This shows the concerted formation of bonds, as expected of a Diels Alder cycloaddition, as defined in the introduction of this section.

===Activation Energy===
The activation energy for the transition state is calculated at 298.15K at AM1 and at a higher level of B3LYP/6-31G* levels respectively for comparison. See table 2.6:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.6: Summary of energies''
|----
|
| colspan="2" | '''AM1'''
| colspan="2" | '''B3LYP/6-31G*'''
|-
|
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies '''
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies'''
|----
|'''Reactants (Cis-Butadiene+Ethene)''' /Hartree
|0.0750
|0.2594
| -234.5991
| -234.4299
|----
|'''Transition State''' /Hartree
|0.1116
|0.2594
| -234.5734
| -234.4876
|----
|'''Activation Energy E<sub>A</sub> /kcal mol<sup>-1</sup>
| --
|22.95
| --
|16.11
|----
|}

To view opt+freq of TS by DFT:{{DOI|10042/to-6230}} <br />
To view opt+freq of cis butadiene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_FREQ-DFT.LOG <br />
To veiw opt+freq of ethene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_OPT%2BFREQ_%28DFT%29.LOG

The energy differences between the transition state and ground state reactant are the activation energy, and only these differences can be compared (but not the absolute energies). It is found that the E<sub>A</sub> found at higher level of theory is smaller. However, both are of relatively similar orders of magnitude.

==Ex 3: Regioselectivity==

In this small exercise, the Diels-Alder cycloaddition between 1,3-Cyclohexadiene and Maleic Anhydride are investigated -- pertaining to its regioselectivity of exo or endo product. From literature, both theoretical and experimental, it is well known that endo-products are preferentially formed as the fast, kinetic product due to transition state stabilisation and thus lowered activation energy.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref><ref name="woodward hoffman">R. Hoffman, R. Woodward, ''J. Chem. Soc.'', 1965, '''87''', 4388-4389: {{DOI|10.1021/ja00947a033}}</ref> This will be verified here via computational predictions.

===Optimisation===
Both the endo and exo transition states are optimised using AM1 and B3LYP/6-31G* repectively, by optimising to TS(Berny). Summary to the calculations are in table 2.7.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.7: Comparison of optimisation of transition states''
|----
|
| colspan="2" | '''Exo-TS'''
| colspan="2" | '''Endo-TS'''
|----
|
|'''AM1'''
|'''B3LYP/6-31G*'''
|'''AM1'''
|'''B3LYP/6-31G*'''
|----
| Jmol
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Exo_TS.mol</uploadedFileContents></jmolApplet></jmol>
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Endo_TS.mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
| -0.0505
| -612.6793
| -0.0516
| -612.6833
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -811
| -448
| -806
| -446
|----
|Vibrations
| colspan="2" | [[Image:YL_Exo_vib.jpg|150px]]
| colspan="2" | [[Image:YL_Endo_vib.jpg|170px]]
|----
|}

To view opt+freq of exo-TS by AM1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:EXO_TS_OPT%2BFREQ.LOG <br />
To view opt+freq of endo-TS by AM1: (file size too big to upload here) <br />
To view opt+freq of exo-TS by DFT:{{DOI|10042/to-6232}}<br />
To view opt+freq of endo-TS by DFT:{{DOI|10042/to-6231}}

As can be seen from above, endo transition state is more stable (i.e. lower in energy) than exo transition state, indicating that endo transition state is favoured – a lower transition state energy leads to reduced activation energy, and a preferred kinetic product.

Imaginary frequencies of exo and endo states show that the bond formation occurs in a concerted, synchronous manner, agreeing with the definition of Diels Alder cycloaddition definition.

===Molecular Orbitals===
HOMO and LUMO orbitals of both exo and endo TS are summaried in table 2.8 below:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.8: HOMO and LUMO of Exo- & Endo- TS''
|----
|
| colspan="2" | '''Exo TS'''
| colspan="2" | '''Endo TS'''
|----
|
|'''HOMO'''
|'''LUMO'''
|'''HOMO'''
|'''LUMO'''
|----
|Structure
|[[Image:YL_Exo_HOMO.jpg|150px]]
|[[Image:YL_Exo_LUMO.jpg|150px]]
|[[Image:YL_Endo_HOMO.jpg|150px]]
|[[Image:YL_Endo_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.3427
| -0.0404
| -0.3451
| -0.0357
|----
|Symmetry wrt Plane
|Antisymmetric
|Antisymmetric
|Antisymmetric
|Antisymmetric
|----
|}

To view calculations on exo-TS MOs: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:EXO_TS_MO.LOG <br />
To view calculations on endo-TS MOs: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ENDO_TS_MO.LOG

All HOMO and LUMO orbitals of exo and endo TS are antisymmetric with respect to the mirror plane. Comparing with the HOMO and LUMOs of starting materials, it is observed that the transition state HOMOs consist of the interaction between HOMO of 1,3-cyclohexadiene and LUMO of maleic anhydride. Hence, this Diels Alder cycloaddition is between the HOMO of the diene and the LUMO of the dienophile.

===Geometries===

[[Image:Exo_geometry_labelled.jpg|380px]] [[Image:Endo_geometry_labelled.jpg|360px]]

Figure on the left: exo-TS. Figure on the right: endo-TS.

Note: All lengths are labelled in Å, all angles in degrees /°. All H atoms are not shown.

Since this is a concerted reaction, the breaking of maleic anhydride/cyclohexadiene bonds and forming of new σ-bonds are expected to be about the same length – which is indeed observed in both exo and endo isomers, where the forming/breaking length occurs at about 1.40Å (1.39 to 1.41Å).

It is noted that the distance between the maleic anhydride and the cyclohexadiene component above it is 2.95Å in exo isomer but 2.89Å in the endo isomer, suggesting that the steric crowding would cause endo-isomer to be less stable. However in contrary endo-TS is in fact more stable, and this phenomenon is explained by the stabilising overlap between π* (C=O) of maleic anhydride and π(C-C) of 1,3-cyclohexadiene, otherwise known as secondary orbital overlap, as demonstrated in the past by e.g. Fox et al <ref>M.A. Fox, R.Cardona, N.J. KIwiet, ''J. Org. Chem.'' 1987, '''52''', 1469-1474.{{DOI|10.1021/jo00384a016}}</ref>. This stabilises the whole system, and hence endo isomer is of a lower energy than its exo counterpart. It is to be noted that secondary orbital overlap occurs on top of the HOMO-LUMO primary orbital interaction of the frontier orbitals. A simplified analogous diagram below, by Fox et al, shows the stabilising interactions with secondary orbital contribution in endo isomer as compared to the exo-isomer.

[[Image:Secondary_orb_exo.jpg|300px]] [[Image:Secondary_orb_endo.jpg|300px]]

Other than these, most of the bond lengths are similar in both structures. More significant differences appear in the newly-formed bond angle between the two components -- 92.7° in exo-TS and 96.7° in endo-TS.

In conclusion, this exercise proves that, in agreement with literature, endo product is favoured in a Diels Alder reaction, due to its transition state being more stabilised (i.e. lower in energy) than its exo isomer, hence endo product is the kinetic product while exo is thermodynamic product.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref>

=Conclusions=

In conclusions, computational methods using Gaussian calculations (with various levels of theory and different approaches) can be used to predict the viability, stability and feasibility of transition states for a pericyclic reaction, eg cycloaddition (Diels-Alder) or sigmatropic Cope rearrangement. Both steric and electronic aspects can be studied by investigating energies, geometries, structures, frontier orbitals, and reaction coordinate/reaction pathways, providing a more holistic understanding of a reaction, especially if it is kinetically controlled by studying its possible transition states.

=References=
<references />

Rep:Mod:mariuspontmercy

2010-12-16T01:20:19Z

Ly308: /* Molecular Orbitals */

'''''Note to Marker''': As most of the calculations are rather fast, they are mostly run on the laptop instead of on the college SCAN platform. As a result, most of these do not have a D-space link to the public repository. The major output files (eg .log or .chk) are hence uploaded as an "image" link and you can access it by clicking it/downloading it -- as per instructions given to us in Module 2''

=Cope Rearrangement=

Cope rearrangement is a percicyclic reaction of [3,3] sigmatropic nature, defined by its characteristic concerted migration of a group from one point of attachment to a conjugated system to another point of attachment. It involves the breaking and formation of one σ-bond each during the process. Here, the reactant, 1,5-hexadiene, is optimised to its minimum-energy (or most stable) conformation before a transition state is calculated. Intrinsic reaction coordinate (IRC) is finally used to relate the starting material/final product comformation to the transitions tate.

==Optimisation of Reactant==
===Conformers of 1,5-hexadiene: HF/3-21G===

1,5-Hexadiene was optimised with an “anti” linkage (i.e. antiperiplanar dihedral angle at its central four carbons), by using Hartree Fork (HF) and a basis set of 3-21G. The ''Anti-1'' conformer was obtained and symmetrised in Gaussview. Next, a gauche conformer (''Gauche3'') was obtained in a similar manner by altering its central dihedral angle. The results are summarized in table 1.1.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.1: Optimisation of 1,5-Hexadiene Conformers''
|----
|
|'''''Anti-1'''''
|'''''Anti-2'''''
|'''''Gauche-3'''''
|----
|Optimised Energy /Hartree
| -231.6926
| -231.6925
| -231.6927
|----
|Optimised Energy /kJ mol<sup>-1</sup>
| -614541
| -614541
| -614542
|----
|Point Group Symmetry
|C<sub>2</sub>
|C<sub>i</sub>
|C<sub>1</sub>
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Hexadiene_anti1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_anti2.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|----
|}

Counter intuitively, the ''gauche3'' conformer is more stable than the ''anti1'', even though the steric crowding in ''gauche3'' could lead to repulsions and destabilisation. Literature results, eg reported by Rocque et al <ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> have also famously proven that gauche conformers are more stable and preferred than anti ones. This is explained by the stereoelectronic effect, which involves stabilisating interaction between the π-electrons of the C=C and vinyl protons, otherwise known as CH-π interaction, thus stabilising ''gauche3'' instead.<ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> This is confirmed by performing another optimisation on an anti-conformer – ''anti2''. And indeed, it is observed that ''anti2'' and ''anti1'' are comparable in their energies, and ''gauche3'' is lower in energy.

To view optimisation files for anti1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI1.LOG

To view optimisation files for anti2: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI.2LOG

To view optimisation files for gauche3: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_GAUCHE3_OPT.LOG

===Optimisation of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

''Anti2'' conformer is optimised by B3LYP method, with basis set of 6-31G*. A summary and comparison of both methods (HF/3-21G and DFT/B3LYP/6-31G*) are shown in table 1.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.2: Comparison of optimisation properties of anti2 conformer, by different methods''
|----
|'''Method'''
|'''HF/3-21G'''
|'''DFT/B3LYP/6-31G*'''
|----
|Optimised Energy/Hartree
| -231.6925
| -234.6117
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614541
| -622284
|----
|'''Geometry'''
|[[Image:Hexadiene_anti2.jpg|400px]]
|[[Image:Hexadiene_anti2.jpg|400px]]
|----
|Bond lengths C1-C2 & C12-C13/Å
|1.33
|1.32
|----
|Bond lengths C2-C6 & C7-C12/Å
|1.50
|1.51
|----
|Bond length C6-C7/Å
|1.55
|1.55
|----
|Bond angles C1-C2-C6 & C7-C12-C13/°
|125.3
|124.8
|----
|Bond angles C2-C6-C7 & C6-C7-C12/°
|112.7
|111.4
|----
|Dihedral C1-C2-C6-C7 & C6-C7-C12-C13/°
|118.5
|114.7
|----
|Dihedral C2-C6-C7-C12/°
|180.0
|180.0
|----
|}

To view DFT optimisation file: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI2_DFT_opt.LOG
To view DFT freq file: {{DOI|10042/to-6215}}

As can be seen, the final energy calculated by DFT is lower than that by HF, and their geometry, though very similar, differs slightly e.g. dihedral angles at the sides, vinyl bond angles, and alkene bond angles.

===Thermochemistry of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

Thermochemistry data is obtained by doing a frequency calculation on the optimised structure, at the same level of theory DFT/B3LYP/6-31G*, at temperatures 0K and 298.15K respectively. Data is summarised in table 1.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.3: Comparison of Thermochemistry Data of Anti2 At 0K & 298.15K''
|----
|
|'''T=0K'''
|'''T=298.15K'''
|----
|Sum of electronic and zero point energies/Hartree
| -234.4663
| -234.4692
|----
|Sum of electronic and thermal energies/Hartree
| -234.4607
| -234.4619
|----
|Sum of electronic and thermal enthalpies/Hartree
| -234.4597
| -234.4609
|----
|Sum of electronic and thermal free energies/Hartree
| -234.4955
| -234.5008
|----
|}

To view opt+freq file at 0K: {{DOI|10042/to-6216}}

It is observed that the energies are higher for the system at 0K than at 298.15K, indicating that the molecule absorbs more heat at higher temperatures and thus thermal contribution to the energies increases.

==''Chair'' Product==

===Optimisation of Chair Product===

Two approaches are used to optimise the chair transition state structure, both at the level of theory of HF/3-21G, at default temperature of 198.15K and 1.0atm pressure. One was done using optimisation to TS(Berny) directly, calculating the force constant only once. The other was done using the redundant coordinate, first to freeze the breaking/forming bond at 2.2Å and optimise to minimum, then finally removing the freeze and optimise the whole structure to a minimum transition state TS(Berny). Results are summaried in table 1.4:

{| class="wikitable" border="1"
|+ ''Table 1.4: Comparison of Optimisation of Chair- Transition State with Different Conditions''
|----
|
|'''TS (Berny)'''<br />{{DOI|10042/to--6217}}
|'''TS (Berny) with Redundant<br />Coordinate''' <br />{{DOI|10042/to--6218}}<br />{{DOI|10042/to--6219}}
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_ts_TSBerny.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_TSBerny_(mod_red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Optimised Energy/Hartree
| -231.6193
| -231.6193
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614347.0313
| -614347.0313
|----
|Bond forming length/Å
|2.02043
|2.02069
|----
|Bond breaking length/Å
|2.02051
|2.02074
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -818
| -818
|----
|Vibration Figure
|[[Image:Chair_ts_HF_freq818.jpg|150px]]
|[[Image:Chair_ts_HF_freq818(mod-red).jpg|150px]]
|----
|}

Both sets of results yielded very similar outcomes, suggesting that either method is reliable and sufficient. Looking at the imaginary frequency at -818cm<sup>-1</sup>, it can be seen that the bond breaking and forming motions occur in a concerted manner simultaneously and asynchronously, corresponding to the definition of pericyclic sigmatropic reaction.

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

Intrinsic reaction coordinate (IRC) calculation was run to track the energy of the transition state along its reaction pathway, reaching a minimum value. The number of points along IRC was set at 100, and a minimum was reached successfully. This calculation is repeated for both force constant calculation for once and for always. See table 1.5

{| class="wikitable" border="1"
|+''Table 1.5: Comparison of IRC results for Chair-TS ''
|----
|'''Force Constant Computation'''
|'''Energy/Hartree'''
|'''Energy along IRC Path'''
|'''Gradient'''
|'''Minimum Structure'''
|----
|'''Once''' <br /> {{DOI|10042/to-6220}}
| -231.6851
|[[Image:Chair_IRC_(ONCE)_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_min_str.jpg|200px]]
|----
|'''Always'''<br /> {{DOI|10042/to-6221}}
| -231.6917
|[[Image:Chair_IRC_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_minimum_structure.jpg|200px]]
|----
|}

For the “force constant=once” calculation, even though IRC points set to 100, only 21 steps were performed by the programme and the energy graph did not plateau to a minimum, and its corresponding RMS gradient definitely did not reach zero. Hence, it did not successfully reach a minimum geometry. However, for the “force constant=always” calculation, the gradient has fallen to zero, it shows that the total energy of the structure has indeed fallen to a plateau minimum successfully, due to force constants being calculated at each and every step. This successful calculation can then be related to the gauche3 conformer obtained above, which has a comparable energy of -231.6927.

==Optimisation of ''Boat'' Product==

The boat transition state is optimised using yet another approach – with QST2 at the same level of theory. Here, it was ensured that the numbering of the atoms in reactant and that in product match exactly; otherwise, the calculation fails. Summary of optimisation is shown in table 1.6:

{| class="wikitable" border="1"
|+''Table 1.6: Optimisation results of Boat-Transition State by QST2''
|----
|Energy /Hartree
| -231.6028
|----
|Energy/kJ mol<sup>-1</sup>
| -614303.2667
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -840
|----
|Bond forming/breaking length/Å
|2.14
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Boat.mol</uploadedFileContents></jmolApplet></jmol>
|----
|D-space link
|{{DOI|10042/to-6223}}
|----
|}

Looking at the imaginary frequency at -840cm<sup>-1</sup>, the vibration motion stretches horizontally in plane in concerted and asynchronous manner, indicating again that the breaking and formation of bond is concerted and simultaneous, confirming the definition of Cope rearrangement.

==Activation Energy==

Thermochemical energies of chair/boat transition states are compared with reactant, and each corresponding activation energies for chair and boat pathways are calculated and compared, across T=0K and T=298.15K, all done at the same level of theory at DFT/B3LYP/6-31G*. The summary of energies are shown in table 1.7:

{| class="wikitable" border="1"
|+''Table 7: Comparison of Activation Energies of Chair & Boat TS at 0K & 298.15K''
|----
|
|
| colspan="4" | '''T=0K'''
| colspan="3" | '''T=298.15K'''
|----
|
| '''Electronic Energy'''
|'''Sum of electronic & zero-point energies/Hartree'''
|'''Activation Energy E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|'''E<sub>A</sub>(expt, 0K)/kcal mol<sup>-1</sup>'''
|'''Sum of electronic & thermal free energies/Hartree'''
|'''E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|----
|'''Chair TS'''
| -231.6193
| -234.4119
|0.0544
|34.11
|33.5+-0.5
| -234.4082
|0.0537
|33.67
|----
|'''Boat TS'''
| -231.6028
| -234.3990
|0.0673
|42.23
|44.7 +- 2.0
| -234.3960
|0.0659
|41.32
|----
|'''Reactant'''
| -231.6925
| -234.4663
|NA
|NA
|NA
| -234.4619
|NA
|NA
|----
|}

E<sub>A</sub> for T=0K is done by taking the difference between the sum of electronic & zero-point energies of reactant anti2 1,5-hexadiene and the chair/boat transition state, followed by converting the units from Hartree to kcal/mol (multiply by 627.509). E(A) for T=298.15K is done by the difference between their sum of electronic & thermal energies instead. The activation energy is found to be always higher for the boat than chair conformation, which agrees with expectation that chair-cyclohexane is usually the more stable form, with minimal steric clash due to the staggered H’s. In boat TS, the eclipsed hydrogens will tend to result in steric or torsional strain, causing destabilisation of the structure.

At T=0K, computational values found here are in good agreement with literature experimental values, with the values being just very slightly out of range from experimental data. This could be due to many trivial causes, especially since computational methods generate slightly different results each time it is run, or it could be due to the degree of accuracy imposed by the level of theory used. However, generally, the trend that chair-TS is more stable than its boat counterpart is well followed by computational methods here.

To view Chair-TS opt+freq by DFT:{{DOI|10042/to-6224}}{{DOI|10042/to-6225}} <br />
To view Boat-TS opt+freq by DFT: {{DOI|10042/to-6222}}{{DOI|10042/to-6226}}

=Diels Alder Reaction=

The Diels-Alder reaction is a [4+2] cycloaddition, between a conjugated diene and an dienophile, involving the concerted formation of two (or more) σ-bonds between the termini of two or more conjugated π-systems. The reaction investigated here is between cis-butadiene and ethylene; and the reaction between 1,3-cyclohexadiene and maleic anhydride is studied for its regioselectivity property.

==Ex 1: HOMO & LUMO of Reactants==

The HOMO and LUMO for reactants (cis-butadiene and ethylene) are obtained upon their optimised structure, done at the level of theory of semi-empirical/AM1. Their properties are summarised in table 2.1:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.1: HOMO & LUMO of cis-Butadiene & Ethylene (Reactants)''
|----
|
|'''Cis-Butadiene'''
|'''Ethylene'''
|----
|'''HOMO'''
|[[Image:YL_Butadiene_HOMO.jpg|150px]]
|[[Image:YL_Ethene_HOMO.jpg|150px]]
|----
|Energy /Hartree
| -0.34381
| -0.38777
|----
|Symmetry
|anti-symmetric
|symmetric
|----
|'''LUMO'''
|[[Image:YL_Butadiene_LUMO.jpg|150px]]
|[[Image:YL_Ehene_LUMO.jpg|150px]]
|----
|Energy /Hartree
|0.01708
|0.05284
|----
|Symmetry
|symmetric
|anti-symmetric
|----
|}

To view cis-butadiene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_MO.LOG <br />
To view ethylene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_MO.LOG

==Ex 2: Transition State Geometries==

===Optimisation/Geometry===

The transition state of the reaction is optimised with TS(Berny) with specified redundant coordinate method – the breaking/forming bonds are frozen to 2.1Å first and optimised to a minimum, before optimising with bond derivative to TS(Berny). The level of theory used is semiempirical/AM1. The optimisation summary shown in table 2.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.2: Optimisation of transition state''
|----
|Structure
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>TS_(mod-red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
|0.11163
|----
|Energy/kJ mol<sup>-1</sup>
|296
|----
|Bond forming length/Å
|2.1189
|----
|Bond breaking length/Å
|2.1196
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -956
|----
|Vibration
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|}

To view optimisation for TS: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:TS_OPT2_MOD-RED.LOG

The optimised geometry is further investigated in table 2.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.3: Geometry of optimised transition state''
|----
|
|'''Transition State'''
|'''Butadiene'''
|'''Ethene'''
|----
|Structure
|[[Image:TS_with_labels.jpg|350px]]
|[[Image:Butadiene_with_labels.jpg|250px]]
|[[Image:Ethene_with_labels.jpg|250px]]
|----
|Bond length C6-C4/Å
|1.4
|1.45
|NA
|----
|Bond length C1-C4/Å
|1.38
|1.34
|NA
|----
|Bond length C6-C8/Å
|1.38
|1.34
|NA
|----
|Bond length C11-C12/Å
|1.38
|NA
|1.33
|----
|Bond angle (C4-C1-C12 & C6-C8-C11)/
|99.3
|
|NA
|----
|Bond forming (C6-C11)/Å
|2.12
|NA
|NA
|----
|Bond breaking (C1-C12)/Å
|2.12
|NA
|NA
|----
|}

From literature <ref name="0031-8949">F.H. Allen, Olga Kennard, D.G. Watson, ''J. Chem. Soc. Perkin Trans. II'', 1987, S1-S18 </ref><ref name="dewarcc">M. J. S. Dewar and H. N. Schmeising, ''Tetrahedron'', 1960, '''11''', {{DOI|10.1016/0040-4020(60)89012-6}} </ref>, a typical sp<sup>3</sup> C-C bond is 1.544Å long, while sp<sup>2</sup> C-C bond is 1.338Å long, and that the van der walls radius of carbon is 1.7Å. Judging from the data obtained, the partially formed bonds (2.12Å) are all longer than typical sp<sup>3</sup> C-C bond but shorter than twice the van der waals radius of carbon. It thus suggests that there is indeed bonding/overlap, and since a bond is still in the process of forming, it is an incomplete C-C bond, and hence, longer than usual.

===Molecular Orbitals===
The HOMO and LUMO orbitals of the transition states are shown in table 2.4:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.4: HOMO and LUMO of transition state''
|----
|
|'''HOMO'''
|'''LUMO'''
|----
|MO
|[[Image:YL_TS_HOMO.jpg|150px]]
|[[Image:YL_TS_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.32392
|0.02315
|----
|Symmetry (wrt plane)
|antisymmetric
|symmetric
|----
|}

Comparing to the HOMO and LUMO orbitals of the reactants shown earlier, it can be seen that the HOMO of transition state consists of the antisymmetric HOMO of cis-butadiene and the antisymmetric LUMO of ethane, forming a π(butadiene, diene)-π*(ethylene, dienophile) interaction, typical of cycloadditions. Since 6π electrons are involved (i.e. 4n+2 where n=1), under thermal conditions it would proceed via the Huckel route with all suprafacial components, and hence the bond forms on the same face of both the butadiene and ethylene π systems.

The HOMO(butadiene)-LUMO(ethylene) interaction is more favourable than the other way round due to the smaller energy gap:
*HOMO(butadiene)-LUMO(ethene) = 0.39665 Hartree (smaller, more favourable)
*HOMO(ethene)-LUMO(butadiene)= 0.40485 Hartree (larger, not favoured)

===Vibrational Frequency===
{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.5: Vibrations of Transition State''
|----
|'''Frequency/cm<sup>-1</sup>'''
|'''Description'''
|'''Mode of Vibration'''
|----
| -956
|Synchronous and concerted bond formation
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|147
|Rocking: ethene unit rocks about butadiene unit.
|[[Image:TS_vib_freq-147.jpg|150px]]
|----
|}

From the imaginary frequency of vibration, it can be seen that both bonds form in a concerted synchronous manner, where the ethene carbons move towards or away from butadiene unit together. This shows the concerted formation of bonds, as expected of a Diels Alder cycloaddition, as defined in the introduction of this section.

===Activation Energy===
The activation energy for the transition state is calculated at 298.15K at AM1 and at a higher level of B3LYP/6-31G* levels respectively for comparison. See table 2.6:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.6: Summary of energies''
|----
|
| colspan="2" | '''AM1'''
| colspan="2" | '''B3LYP/6-31G*'''
|-
|
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies '''
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies'''
|----
|'''Reactants (Cis-Butadiene+Ethene)''' /Hartree
|0.0750
|0.2594
| -234.5991
| -234.4299
|----
|'''Transition State''' /Hartree
|0.1116
|0.2594
| -234.5734
| -234.4876
|----
|'''Activation Energy E<sub>A</sub> /kcal mol<sup>-1</sup>
| --
|22.95
| --
|16.11
|----
|}

To view opt+freq of TS by DFT:{{DOI|10042/to-6230}} <br />
To view opt+freq of cis butadiene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_FREQ-DFT.LOG <br />
To veiw opt+freq of ethene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_OPT%2BFREQ_%28DFT%29.LOG

The energy differences between the transition state and ground state reactant are the activation energy, and only these differences can be compared (but not the absolute energies). It is found that the E<sub>A</sub> found at higher level of theory is smaller. However, both are of relatively similar orders of magnitude.

==Ex 3: Regioselectivity==

In this small exercise, the Diels-Alder cycloaddition between 1,3-Cyclohexadiene and Maleic Anhydride are investigated -- pertaining to its regioselectivity of exo or endo product. From literature, both theoretical and experimental, it is well known that endo-products are preferentially formed as the fast, kinetic product due to transition state stabilisation and thus lowered activation energy.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref><ref name="woodward hoffman">R. Hoffman, R. Woodward, ''J. Chem. Soc.'', 1965, '''87''', 4388-4389: {{DOI|10.1021/ja00947a033}}</ref> This will be verified here via computational predictions.

===Optimisation===
Both the endo and exo transition states are optimised using AM1 and B3LYP/6-31G* repectively, by optimising to TS(Berny). Summary to the calculations are in table 2.7.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.7: Comparison of optimisation of transition states''
|----
|
| colspan="2" | '''Exo-TS'''
| colspan="2" | '''Endo-TS'''
|----
|
|'''AM1'''
|'''B3LYP/6-31G*'''
|'''AM1'''
|'''B3LYP/6-31G*'''
|----
| Jmol
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Exo_TS.mol</uploadedFileContents></jmolApplet></jmol>
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Endo_TS.mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
| -0.0505
| -612.6793
| -0.0516
| -612.6833
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -811
| -448
| -806
| -446
|----
|Vibrations
| colspan="2" | [[Image:YL_Exo_vib.jpg|150px]]
| colspan="2" | [[Image:YL_Endo_vib.jpg|170px]]
|----
|}

To view opt+freq of exo-TS by AM1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:EXO_TS_OPT%2BFREQ.LOG <br />
To view opt+freq of endo-TS by AM1: (file size too big to upload here) <br />
To view opt+freq of exo-TS by DFT:{{DOI|10042/to-6232}}<br />
To view opt+freq of endo-TS by DFT:{{DOI|10042/to-6231}}

As can be seen from above, endo transition state is more stable (i.e. lower in energy) than exo transition state, indicating that endo transition state is favoured – a lower transition state energy leads to reduced activation energy, and a preferred kinetic product.

Imaginary frequencies of exo and endo states show that the bond formation occurs in a concerted, synchronous manner, agreeing with the definition of Diels Alder cycloaddition definition.

===Molecular Orbitals===
HOMO and LUMO orbitals of both exo and endo TS are summaried in table 2.8 below:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.8: HOMO and LUMO of Exo- & Endo- TS''
|----
|
| colspan="2" | '''Exo TS'''
| colspan="2" | '''Endo TS'''
|----
|
|'''HOMO'''
|'''LUMO'''
|'''HOMO'''
|'''LUMO'''
|----
|Structure
|[[Image:YL_Exo_HOMO.jpg|150px]]
|[[Image:YL_Exo_LUMO.jpg|150px]]
|[[Image:YL_Endo_HOMO.jpg|150px]]
|[[Image:YL_Endo_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.3427
| -0.0404
| -0.3451
| -0.0357
|----
|Symmetry wrt Plane
|Antisymmetric
|Antisymmetric
|Antisymmetric
|Antisymmetric
|----
|}

To view calculations on exo-TS MOs: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:EXO_TS_MO.LOG <br />
To view calculations on endo-TS MOs: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ENDO_TS_MO.LOG

All HOMO and LUMO orbitals of exo and endo TS are antisymmetric with respect to the mirror plane. Comparing with the HOMO and LUMOs of starting materials, it is observed that the transition state HOMOs consist of the interaction between HOMO of 1,3-cyclohexadiene and LUMO of maleic anhydride. Hence, this Diels Alder cycloaddition is between the HOMO of the diene and the LUMO of the dienophile.

===Geometries===

[[Image:Exo_geometry_labelled.jpg|380px]] [[Image:Endo_geometry_labelled.jpg|360px]]

Figure on the left: exo-TS. Figure on the right: endo-TS.

Note: All lengths are labelled in Å, all angles in degrees /°. All H atoms are not shown.

Since this is a concerted reaction, the breaking of maleic anhydride/cyclohexadiene bonds and forming of new σ-bonds are expected to be about the same length – which is indeed observed in both exo and endo isomers, where the forming/breaking length occurs at about 1.40Å (1.39 to 1.41Å).

It is noted that the distance between the maleic anhydride and the cyclohexadiene component above it is 2.95Å in exo isomer but 2.89Å in the endo isomer, suggesting that the steric crowding would cause endo-isomer to be less stable. However in contrary endo-TS is in fact more stable, and this phenomenon is explained by the stabilising overlap between π* (C=O) of maleic anhydride and π(C-C) of 1,3-cyclohexadiene, otherwise known as secondary orbital overlap, as demonstrated in the past by e.g. Fox et al <ref>M.A. Fox, R.Cardona, N.J. KIwiet, ''J. Org. Chem.'' 1987, '''52''', 1469-1474.{{DOI|10.1021/jo00384a016}}</ref>. This stabilises the whole system, and hence endo isomer is of a lower energy than its exo counterpart. It is to be noted that secondary orbital overlap occurs on top of the HOMO-LUMO primary orbital interaction of the frontier orbitals. A simplified analogous diagram below, by Fox et al, shows the stabilising interactions with secondary orbital contribution in endo isomer as compared to the exo-isomer.

[[Image:Secondary_orb_exo.jpg|300px]] [[Image:Secondary_orb_endo.jpg|300px]]

Other than these, most of the bond lengths are similar in both structures. More significant differences appear in the newly-formed bond angle between the two components -- 92.7° in exo-TS and 96.7° in endo-TS.

In conclusion, this exercise proves that, in agreement with literature, endo product is favoured in a Diels Alder reaction, due to its transition state being more stabilised (i.e. lower in energy) than its exo isomer, hence endo product is the kinetic product while exo is thermodynamic product.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref>

=Conclusions=

In conclusions, computational methods using Gaussian calculations (with various levels of theory and different approaches) can be used to predict the viability, stability and feasibility of transition states for a pericyclic reaction, eg cycloaddition (Diels-Alder) or sigmatropic Cope rearrangement. Both steric and electronic aspects can be studied by investigating energies, geometries, structures, frontier orbitals, and reaction coordinate/reaction pathways, providing a more holistic understanding of a reaction, especially if it is kinetically controlled by studying its possible transition states.

=References=
<references />

File:ENDO TS MO.LOG

2010-12-16T01:19:57Z

Ly308:

File:EXO TS MO.LOG

2010-12-16T01:19:45Z

Ly308:

Rep:Mod:mariuspontmercy

2010-12-16T01:18:04Z

Ly308: /* Optimisation */

'''''Note to Marker''': As most of the calculations are rather fast, they are mostly run on the laptop instead of on the college SCAN platform. As a result, most of these do not have a D-space link to the public repository. The major output files (eg .log or .chk) are hence uploaded as an "image" link and you can access it by clicking it/downloading it -- as per instructions given to us in Module 2''

=Cope Rearrangement=

Cope rearrangement is a percicyclic reaction of [3,3] sigmatropic nature, defined by its characteristic concerted migration of a group from one point of attachment to a conjugated system to another point of attachment. It involves the breaking and formation of one σ-bond each during the process. Here, the reactant, 1,5-hexadiene, is optimised to its minimum-energy (or most stable) conformation before a transition state is calculated. Intrinsic reaction coordinate (IRC) is finally used to relate the starting material/final product comformation to the transitions tate.

==Optimisation of Reactant==
===Conformers of 1,5-hexadiene: HF/3-21G===

1,5-Hexadiene was optimised with an “anti” linkage (i.e. antiperiplanar dihedral angle at its central four carbons), by using Hartree Fork (HF) and a basis set of 3-21G. The ''Anti-1'' conformer was obtained and symmetrised in Gaussview. Next, a gauche conformer (''Gauche3'') was obtained in a similar manner by altering its central dihedral angle. The results are summarized in table 1.1.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.1: Optimisation of 1,5-Hexadiene Conformers''
|----
|
|'''''Anti-1'''''
|'''''Anti-2'''''
|'''''Gauche-3'''''
|----
|Optimised Energy /Hartree
| -231.6926
| -231.6925
| -231.6927
|----
|Optimised Energy /kJ mol<sup>-1</sup>
| -614541
| -614541
| -614542
|----
|Point Group Symmetry
|C<sub>2</sub>
|C<sub>i</sub>
|C<sub>1</sub>
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Hexadiene_anti1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_anti2.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|----
|}

Counter intuitively, the ''gauche3'' conformer is more stable than the ''anti1'', even though the steric crowding in ''gauche3'' could lead to repulsions and destabilisation. Literature results, eg reported by Rocque et al <ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> have also famously proven that gauche conformers are more stable and preferred than anti ones. This is explained by the stereoelectronic effect, which involves stabilisating interaction between the π-electrons of the C=C and vinyl protons, otherwise known as CH-π interaction, thus stabilising ''gauche3'' instead.<ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> This is confirmed by performing another optimisation on an anti-conformer – ''anti2''. And indeed, it is observed that ''anti2'' and ''anti1'' are comparable in their energies, and ''gauche3'' is lower in energy.

To view optimisation files for anti1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI1.LOG

To view optimisation files for anti2: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI.2LOG

To view optimisation files for gauche3: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_GAUCHE3_OPT.LOG

===Optimisation of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

''Anti2'' conformer is optimised by B3LYP method, with basis set of 6-31G*. A summary and comparison of both methods (HF/3-21G and DFT/B3LYP/6-31G*) are shown in table 1.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.2: Comparison of optimisation properties of anti2 conformer, by different methods''
|----
|'''Method'''
|'''HF/3-21G'''
|'''DFT/B3LYP/6-31G*'''
|----
|Optimised Energy/Hartree
| -231.6925
| -234.6117
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614541
| -622284
|----
|'''Geometry'''
|[[Image:Hexadiene_anti2.jpg|400px]]
|[[Image:Hexadiene_anti2.jpg|400px]]
|----
|Bond lengths C1-C2 & C12-C13/Å
|1.33
|1.32
|----
|Bond lengths C2-C6 & C7-C12/Å
|1.50
|1.51
|----
|Bond length C6-C7/Å
|1.55
|1.55
|----
|Bond angles C1-C2-C6 & C7-C12-C13/°
|125.3
|124.8
|----
|Bond angles C2-C6-C7 & C6-C7-C12/°
|112.7
|111.4
|----
|Dihedral C1-C2-C6-C7 & C6-C7-C12-C13/°
|118.5
|114.7
|----
|Dihedral C2-C6-C7-C12/°
|180.0
|180.0
|----
|}

To view DFT optimisation file: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI2_DFT_opt.LOG
To view DFT freq file: {{DOI|10042/to-6215}}

As can be seen, the final energy calculated by DFT is lower than that by HF, and their geometry, though very similar, differs slightly e.g. dihedral angles at the sides, vinyl bond angles, and alkene bond angles.

===Thermochemistry of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

Thermochemistry data is obtained by doing a frequency calculation on the optimised structure, at the same level of theory DFT/B3LYP/6-31G*, at temperatures 0K and 298.15K respectively. Data is summarised in table 1.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.3: Comparison of Thermochemistry Data of Anti2 At 0K & 298.15K''
|----
|
|'''T=0K'''
|'''T=298.15K'''
|----
|Sum of electronic and zero point energies/Hartree
| -234.4663
| -234.4692
|----
|Sum of electronic and thermal energies/Hartree
| -234.4607
| -234.4619
|----
|Sum of electronic and thermal enthalpies/Hartree
| -234.4597
| -234.4609
|----
|Sum of electronic and thermal free energies/Hartree
| -234.4955
| -234.5008
|----
|}

To view opt+freq file at 0K: {{DOI|10042/to-6216}}

It is observed that the energies are higher for the system at 0K than at 298.15K, indicating that the molecule absorbs more heat at higher temperatures and thus thermal contribution to the energies increases.

==''Chair'' Product==

===Optimisation of Chair Product===

Two approaches are used to optimise the chair transition state structure, both at the level of theory of HF/3-21G, at default temperature of 198.15K and 1.0atm pressure. One was done using optimisation to TS(Berny) directly, calculating the force constant only once. The other was done using the redundant coordinate, first to freeze the breaking/forming bond at 2.2Å and optimise to minimum, then finally removing the freeze and optimise the whole structure to a minimum transition state TS(Berny). Results are summaried in table 1.4:

{| class="wikitable" border="1"
|+ ''Table 1.4: Comparison of Optimisation of Chair- Transition State with Different Conditions''
|----
|
|'''TS (Berny)'''<br />{{DOI|10042/to--6217}}
|'''TS (Berny) with Redundant<br />Coordinate''' <br />{{DOI|10042/to--6218}}<br />{{DOI|10042/to--6219}}
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_ts_TSBerny.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_TSBerny_(mod_red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Optimised Energy/Hartree
| -231.6193
| -231.6193
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614347.0313
| -614347.0313
|----
|Bond forming length/Å
|2.02043
|2.02069
|----
|Bond breaking length/Å
|2.02051
|2.02074
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -818
| -818
|----
|Vibration Figure
|[[Image:Chair_ts_HF_freq818.jpg|150px]]
|[[Image:Chair_ts_HF_freq818(mod-red).jpg|150px]]
|----
|}

Both sets of results yielded very similar outcomes, suggesting that either method is reliable and sufficient. Looking at the imaginary frequency at -818cm<sup>-1</sup>, it can be seen that the bond breaking and forming motions occur in a concerted manner simultaneously and asynchronously, corresponding to the definition of pericyclic sigmatropic reaction.

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

Intrinsic reaction coordinate (IRC) calculation was run to track the energy of the transition state along its reaction pathway, reaching a minimum value. The number of points along IRC was set at 100, and a minimum was reached successfully. This calculation is repeated for both force constant calculation for once and for always. See table 1.5

{| class="wikitable" border="1"
|+''Table 1.5: Comparison of IRC results for Chair-TS ''
|----
|'''Force Constant Computation'''
|'''Energy/Hartree'''
|'''Energy along IRC Path'''
|'''Gradient'''
|'''Minimum Structure'''
|----
|'''Once''' <br /> {{DOI|10042/to-6220}}
| -231.6851
|[[Image:Chair_IRC_(ONCE)_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_min_str.jpg|200px]]
|----
|'''Always'''<br /> {{DOI|10042/to-6221}}
| -231.6917
|[[Image:Chair_IRC_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_minimum_structure.jpg|200px]]
|----
|}

For the “force constant=once” calculation, even though IRC points set to 100, only 21 steps were performed by the programme and the energy graph did not plateau to a minimum, and its corresponding RMS gradient definitely did not reach zero. Hence, it did not successfully reach a minimum geometry. However, for the “force constant=always” calculation, the gradient has fallen to zero, it shows that the total energy of the structure has indeed fallen to a plateau minimum successfully, due to force constants being calculated at each and every step. This successful calculation can then be related to the gauche3 conformer obtained above, which has a comparable energy of -231.6927.

==Optimisation of ''Boat'' Product==

The boat transition state is optimised using yet another approach – with QST2 at the same level of theory. Here, it was ensured that the numbering of the atoms in reactant and that in product match exactly; otherwise, the calculation fails. Summary of optimisation is shown in table 1.6:

{| class="wikitable" border="1"
|+''Table 1.6: Optimisation results of Boat-Transition State by QST2''
|----
|Energy /Hartree
| -231.6028
|----
|Energy/kJ mol<sup>-1</sup>
| -614303.2667
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -840
|----
|Bond forming/breaking length/Å
|2.14
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Boat.mol</uploadedFileContents></jmolApplet></jmol>
|----
|D-space link
|{{DOI|10042/to-6223}}
|----
|}

Looking at the imaginary frequency at -840cm<sup>-1</sup>, the vibration motion stretches horizontally in plane in concerted and asynchronous manner, indicating again that the breaking and formation of bond is concerted and simultaneous, confirming the definition of Cope rearrangement.

==Activation Energy==

Thermochemical energies of chair/boat transition states are compared with reactant, and each corresponding activation energies for chair and boat pathways are calculated and compared, across T=0K and T=298.15K, all done at the same level of theory at DFT/B3LYP/6-31G*. The summary of energies are shown in table 1.7:

{| class="wikitable" border="1"
|+''Table 7: Comparison of Activation Energies of Chair & Boat TS at 0K & 298.15K''
|----
|
|
| colspan="4" | '''T=0K'''
| colspan="3" | '''T=298.15K'''
|----
|
| '''Electronic Energy'''
|'''Sum of electronic & zero-point energies/Hartree'''
|'''Activation Energy E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|'''E<sub>A</sub>(expt, 0K)/kcal mol<sup>-1</sup>'''
|'''Sum of electronic & thermal free energies/Hartree'''
|'''E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|----
|'''Chair TS'''
| -231.6193
| -234.4119
|0.0544
|34.11
|33.5+-0.5
| -234.4082
|0.0537
|33.67
|----
|'''Boat TS'''
| -231.6028
| -234.3990
|0.0673
|42.23
|44.7 +- 2.0
| -234.3960
|0.0659
|41.32
|----
|'''Reactant'''
| -231.6925
| -234.4663
|NA
|NA
|NA
| -234.4619
|NA
|NA
|----
|}

E<sub>A</sub> for T=0K is done by taking the difference between the sum of electronic & zero-point energies of reactant anti2 1,5-hexadiene and the chair/boat transition state, followed by converting the units from Hartree to kcal/mol (multiply by 627.509). E(A) for T=298.15K is done by the difference between their sum of electronic & thermal energies instead. The activation energy is found to be always higher for the boat than chair conformation, which agrees with expectation that chair-cyclohexane is usually the more stable form, with minimal steric clash due to the staggered H’s. In boat TS, the eclipsed hydrogens will tend to result in steric or torsional strain, causing destabilisation of the structure.

At T=0K, computational values found here are in good agreement with literature experimental values, with the values being just very slightly out of range from experimental data. This could be due to many trivial causes, especially since computational methods generate slightly different results each time it is run, or it could be due to the degree of accuracy imposed by the level of theory used. However, generally, the trend that chair-TS is more stable than its boat counterpart is well followed by computational methods here.

To view Chair-TS opt+freq by DFT:{{DOI|10042/to-6224}}{{DOI|10042/to-6225}} <br />
To view Boat-TS opt+freq by DFT: {{DOI|10042/to-6222}}{{DOI|10042/to-6226}}

=Diels Alder Reaction=

The Diels-Alder reaction is a [4+2] cycloaddition, between a conjugated diene and an dienophile, involving the concerted formation of two (or more) σ-bonds between the termini of two or more conjugated π-systems. The reaction investigated here is between cis-butadiene and ethylene; and the reaction between 1,3-cyclohexadiene and maleic anhydride is studied for its regioselectivity property.

==Ex 1: HOMO & LUMO of Reactants==

The HOMO and LUMO for reactants (cis-butadiene and ethylene) are obtained upon their optimised structure, done at the level of theory of semi-empirical/AM1. Their properties are summarised in table 2.1:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.1: HOMO & LUMO of cis-Butadiene & Ethylene (Reactants)''
|----
|
|'''Cis-Butadiene'''
|'''Ethylene'''
|----
|'''HOMO'''
|[[Image:YL_Butadiene_HOMO.jpg|150px]]
|[[Image:YL_Ethene_HOMO.jpg|150px]]
|----
|Energy /Hartree
| -0.34381
| -0.38777
|----
|Symmetry
|anti-symmetric
|symmetric
|----
|'''LUMO'''
|[[Image:YL_Butadiene_LUMO.jpg|150px]]
|[[Image:YL_Ehene_LUMO.jpg|150px]]
|----
|Energy /Hartree
|0.01708
|0.05284
|----
|Symmetry
|symmetric
|anti-symmetric
|----
|}

To view cis-butadiene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_MO.LOG <br />
To view ethylene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_MO.LOG

==Ex 2: Transition State Geometries==

===Optimisation/Geometry===

The transition state of the reaction is optimised with TS(Berny) with specified redundant coordinate method – the breaking/forming bonds are frozen to 2.1Å first and optimised to a minimum, before optimising with bond derivative to TS(Berny). The level of theory used is semiempirical/AM1. The optimisation summary shown in table 2.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.2: Optimisation of transition state''
|----
|Structure
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>TS_(mod-red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
|0.11163
|----
|Energy/kJ mol<sup>-1</sup>
|296
|----
|Bond forming length/Å
|2.1189
|----
|Bond breaking length/Å
|2.1196
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -956
|----
|Vibration
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|}

To view optimisation for TS: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:TS_OPT2_MOD-RED.LOG

The optimised geometry is further investigated in table 2.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.3: Geometry of optimised transition state''
|----
|
|'''Transition State'''
|'''Butadiene'''
|'''Ethene'''
|----
|Structure
|[[Image:TS_with_labels.jpg|350px]]
|[[Image:Butadiene_with_labels.jpg|250px]]
|[[Image:Ethene_with_labels.jpg|250px]]
|----
|Bond length C6-C4/Å
|1.4
|1.45
|NA
|----
|Bond length C1-C4/Å
|1.38
|1.34
|NA
|----
|Bond length C6-C8/Å
|1.38
|1.34
|NA
|----
|Bond length C11-C12/Å
|1.38
|NA
|1.33
|----
|Bond angle (C4-C1-C12 & C6-C8-C11)/
|99.3
|
|NA
|----
|Bond forming (C6-C11)/Å
|2.12
|NA
|NA
|----
|Bond breaking (C1-C12)/Å
|2.12
|NA
|NA
|----
|}

From literature <ref name="0031-8949">F.H. Allen, Olga Kennard, D.G. Watson, ''J. Chem. Soc. Perkin Trans. II'', 1987, S1-S18 </ref><ref name="dewarcc">M. J. S. Dewar and H. N. Schmeising, ''Tetrahedron'', 1960, '''11''', {{DOI|10.1016/0040-4020(60)89012-6}} </ref>, a typical sp<sup>3</sup> C-C bond is 1.544Å long, while sp<sup>2</sup> C-C bond is 1.338Å long, and that the van der walls radius of carbon is 1.7Å. Judging from the data obtained, the partially formed bonds (2.12Å) are all longer than typical sp<sup>3</sup> C-C bond but shorter than twice the van der waals radius of carbon. It thus suggests that there is indeed bonding/overlap, and since a bond is still in the process of forming, it is an incomplete C-C bond, and hence, longer than usual.

===Molecular Orbitals===
The HOMO and LUMO orbitals of the transition states are shown in table 2.4:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.4: HOMO and LUMO of transition state''
|----
|
|'''HOMO'''
|'''LUMO'''
|----
|MO
|[[Image:YL_TS_HOMO.jpg|150px]]
|[[Image:YL_TS_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.32392
|0.02315
|----
|Symmetry (wrt plane)
|antisymmetric
|symmetric
|----
|}

Comparing to the HOMO and LUMO orbitals of the reactants shown earlier, it can be seen that the HOMO of transition state consists of the antisymmetric HOMO of cis-butadiene and the antisymmetric LUMO of ethane, forming a π(butadiene, diene)-π*(ethylene, dienophile) interaction, typical of cycloadditions. Since 6π electrons are involved (i.e. 4n+2 where n=1), under thermal conditions it would proceed via the Huckel route with all suprafacial components, and hence the bond forms on the same face of both the butadiene and ethylene π systems.

The HOMO(butadiene)-LUMO(ethylene) interaction is more favourable than the other way round due to the smaller energy gap:
*HOMO(butadiene)-LUMO(ethene) = 0.39665 Hartree (smaller, more favourable)
*HOMO(ethene)-LUMO(butadiene)= 0.40485 Hartree (larger, not favoured)

===Vibrational Frequency===
{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.5: Vibrations of Transition State''
|----
|'''Frequency/cm<sup>-1</sup>'''
|'''Description'''
|'''Mode of Vibration'''
|----
| -956
|Synchronous and concerted bond formation
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|147
|Rocking: ethene unit rocks about butadiene unit.
|[[Image:TS_vib_freq-147.jpg|150px]]
|----
|}

From the imaginary frequency of vibration, it can be seen that both bonds form in a concerted synchronous manner, where the ethene carbons move towards or away from butadiene unit together. This shows the concerted formation of bonds, as expected of a Diels Alder cycloaddition, as defined in the introduction of this section.

===Activation Energy===
The activation energy for the transition state is calculated at 298.15K at AM1 and at a higher level of B3LYP/6-31G* levels respectively for comparison. See table 2.6:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.6: Summary of energies''
|----
|
| colspan="2" | '''AM1'''
| colspan="2" | '''B3LYP/6-31G*'''
|-
|
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies '''
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies'''
|----
|'''Reactants (Cis-Butadiene+Ethene)''' /Hartree
|0.0750
|0.2594
| -234.5991
| -234.4299
|----
|'''Transition State''' /Hartree
|0.1116
|0.2594
| -234.5734
| -234.4876
|----
|'''Activation Energy E<sub>A</sub> /kcal mol<sup>-1</sup>
| --
|22.95
| --
|16.11
|----
|}

To view opt+freq of TS by DFT:{{DOI|10042/to-6230}} <br />
To view opt+freq of cis butadiene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_FREQ-DFT.LOG <br />
To veiw opt+freq of ethene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_OPT%2BFREQ_%28DFT%29.LOG

The energy differences between the transition state and ground state reactant are the activation energy, and only these differences can be compared (but not the absolute energies). It is found that the E<sub>A</sub> found at higher level of theory is smaller. However, both are of relatively similar orders of magnitude.

==Ex 3: Regioselectivity==

In this small exercise, the Diels-Alder cycloaddition between 1,3-Cyclohexadiene and Maleic Anhydride are investigated -- pertaining to its regioselectivity of exo or endo product. From literature, both theoretical and experimental, it is well known that endo-products are preferentially formed as the fast, kinetic product due to transition state stabilisation and thus lowered activation energy.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref><ref name="woodward hoffman">R. Hoffman, R. Woodward, ''J. Chem. Soc.'', 1965, '''87''', 4388-4389: {{DOI|10.1021/ja00947a033}}</ref> This will be verified here via computational predictions.

===Optimisation===
Both the endo and exo transition states are optimised using AM1 and B3LYP/6-31G* repectively, by optimising to TS(Berny). Summary to the calculations are in table 2.7.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.7: Comparison of optimisation of transition states''
|----
|
| colspan="2" | '''Exo-TS'''
| colspan="2" | '''Endo-TS'''
|----
|
|'''AM1'''
|'''B3LYP/6-31G*'''
|'''AM1'''
|'''B3LYP/6-31G*'''
|----
| Jmol
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Exo_TS.mol</uploadedFileContents></jmolApplet></jmol>
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Endo_TS.mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
| -0.0505
| -612.6793
| -0.0516
| -612.6833
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -811
| -448
| -806
| -446
|----
|Vibrations
| colspan="2" | [[Image:YL_Exo_vib.jpg|150px]]
| colspan="2" | [[Image:YL_Endo_vib.jpg|170px]]
|----
|}

To view opt+freq of exo-TS by AM1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:EXO_TS_OPT%2BFREQ.LOG <br />
To view opt+freq of endo-TS by AM1: (file size too big to upload here) <br />
To view opt+freq of exo-TS by DFT:{{DOI|10042/to-6232}}<br />
To view opt+freq of endo-TS by DFT:{{DOI|10042/to-6231}}

As can be seen from above, endo transition state is more stable (i.e. lower in energy) than exo transition state, indicating that endo transition state is favoured – a lower transition state energy leads to reduced activation energy, and a preferred kinetic product.

Imaginary frequencies of exo and endo states show that the bond formation occurs in a concerted, synchronous manner, agreeing with the definition of Diels Alder cycloaddition definition.

===Molecular Orbitals===
HOMO and LUMO orbitals of both exo and endo TS are summaried in table 2.8 below:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.8: HOMO and LUMO of Exo- & Endo- TS''
|----
|
| colspan="2" | '''Exo TS'''
| colspan="2" | '''Endo TS'''
|----
|
|'''HOMO'''
|'''LUMO'''
|'''HOMO'''
|'''LUMO'''
|----
|Structure
|[[Image:YL_Exo_HOMO.jpg|150px]]
|[[Image:YL_Exo_LUMO.jpg|150px]]
|[[Image:YL_Endo_HOMO.jpg|150px]]
|[[Image:YL_Endo_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.3427
| -0.0404
| -0.3451
| -0.0357
|----
|Symmetry wrt Plane
|Antisymmetric
|Antisymmetric
|Antisymmetric
|Antisymmetric
|----
|}

All HOMO and LUMO orbitals of exo and endo TS are antisymmetric with respect to the mirror plane. Comparing with the HOMO and LUMOs of starting materials, it is observed that the transition state HOMOs consist of the interaction between HOMO of 1,3-cyclohexadiene and LUMO of maleic anhydride. Hence, this Diels Alder cycloaddition is between the HOMO of the diene and the LUMO of the dienophile.

===Geometries===

[[Image:Exo_geometry_labelled.jpg|380px]] [[Image:Endo_geometry_labelled.jpg|360px]]

Figure on the left: exo-TS. Figure on the right: endo-TS.

Note: All lengths are labelled in Å, all angles in degrees /°. All H atoms are not shown.

Since this is a concerted reaction, the breaking of maleic anhydride/cyclohexadiene bonds and forming of new σ-bonds are expected to be about the same length – which is indeed observed in both exo and endo isomers, where the forming/breaking length occurs at about 1.40Å (1.39 to 1.41Å).

It is noted that the distance between the maleic anhydride and the cyclohexadiene component above it is 2.95Å in exo isomer but 2.89Å in the endo isomer, suggesting that the steric crowding would cause endo-isomer to be less stable. However in contrary endo-TS is in fact more stable, and this phenomenon is explained by the stabilising overlap between π* (C=O) of maleic anhydride and π(C-C) of 1,3-cyclohexadiene, otherwise known as secondary orbital overlap, as demonstrated in the past by e.g. Fox et al <ref>M.A. Fox, R.Cardona, N.J. KIwiet, ''J. Org. Chem.'' 1987, '''52''', 1469-1474.{{DOI|10.1021/jo00384a016}}</ref>. This stabilises the whole system, and hence endo isomer is of a lower energy than its exo counterpart. It is to be noted that secondary orbital overlap occurs on top of the HOMO-LUMO primary orbital interaction of the frontier orbitals. A simplified analogous diagram below, by Fox et al, shows the stabilising interactions with secondary orbital contribution in endo isomer as compared to the exo-isomer.

[[Image:Secondary_orb_exo.jpg|300px]] [[Image:Secondary_orb_endo.jpg|300px]]

Other than these, most of the bond lengths are similar in both structures. More significant differences appear in the newly-formed bond angle between the two components -- 92.7° in exo-TS and 96.7° in endo-TS.

In conclusion, this exercise proves that, in agreement with literature, endo product is favoured in a Diels Alder reaction, due to its transition state being more stabilised (i.e. lower in energy) than its exo isomer, hence endo product is the kinetic product while exo is thermodynamic product.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref>

=Conclusions=

In conclusions, computational methods using Gaussian calculations (with various levels of theory and different approaches) can be used to predict the viability, stability and feasibility of transition states for a pericyclic reaction, eg cycloaddition (Diels-Alder) or sigmatropic Cope rearrangement. Both steric and electronic aspects can be studied by investigating energies, geometries, structures, frontier orbitals, and reaction coordinate/reaction pathways, providing a more holistic understanding of a reaction, especially if it is kinetically controlled by studying its possible transition states.

=References=
<references />

Rep:Mod:mariuspontmercy

2010-12-16T01:17:28Z

Ly308: /* Optimisation */

'''''Note to Marker''': As most of the calculations are rather fast, they are mostly run on the laptop instead of on the college SCAN platform. As a result, most of these do not have a D-space link to the public repository. The major output files (eg .log or .chk) are hence uploaded as an "image" link and you can access it by clicking it/downloading it -- as per instructions given to us in Module 2''

=Cope Rearrangement=

Cope rearrangement is a percicyclic reaction of [3,3] sigmatropic nature, defined by its characteristic concerted migration of a group from one point of attachment to a conjugated system to another point of attachment. It involves the breaking and formation of one σ-bond each during the process. Here, the reactant, 1,5-hexadiene, is optimised to its minimum-energy (or most stable) conformation before a transition state is calculated. Intrinsic reaction coordinate (IRC) is finally used to relate the starting material/final product comformation to the transitions tate.

==Optimisation of Reactant==
===Conformers of 1,5-hexadiene: HF/3-21G===

1,5-Hexadiene was optimised with an “anti” linkage (i.e. antiperiplanar dihedral angle at its central four carbons), by using Hartree Fork (HF) and a basis set of 3-21G. The ''Anti-1'' conformer was obtained and symmetrised in Gaussview. Next, a gauche conformer (''Gauche3'') was obtained in a similar manner by altering its central dihedral angle. The results are summarized in table 1.1.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.1: Optimisation of 1,5-Hexadiene Conformers''
|----
|
|'''''Anti-1'''''
|'''''Anti-2'''''
|'''''Gauche-3'''''
|----
|Optimised Energy /Hartree
| -231.6926
| -231.6925
| -231.6927
|----
|Optimised Energy /kJ mol<sup>-1</sup>
| -614541
| -614541
| -614542
|----
|Point Group Symmetry
|C<sub>2</sub>
|C<sub>i</sub>
|C<sub>1</sub>
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Hexadiene_anti1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_anti2.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|----
|}

Counter intuitively, the ''gauche3'' conformer is more stable than the ''anti1'', even though the steric crowding in ''gauche3'' could lead to repulsions and destabilisation. Literature results, eg reported by Rocque et al <ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> have also famously proven that gauche conformers are more stable and preferred than anti ones. This is explained by the stereoelectronic effect, which involves stabilisating interaction between the π-electrons of the C=C and vinyl protons, otherwise known as CH-π interaction, thus stabilising ''gauche3'' instead.<ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> This is confirmed by performing another optimisation on an anti-conformer – ''anti2''. And indeed, it is observed that ''anti2'' and ''anti1'' are comparable in their energies, and ''gauche3'' is lower in energy.

To view optimisation files for anti1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI1.LOG

To view optimisation files for anti2: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI.2LOG

To view optimisation files for gauche3: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_GAUCHE3_OPT.LOG

===Optimisation of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

''Anti2'' conformer is optimised by B3LYP method, with basis set of 6-31G*. A summary and comparison of both methods (HF/3-21G and DFT/B3LYP/6-31G*) are shown in table 1.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.2: Comparison of optimisation properties of anti2 conformer, by different methods''
|----
|'''Method'''
|'''HF/3-21G'''
|'''DFT/B3LYP/6-31G*'''
|----
|Optimised Energy/Hartree
| -231.6925
| -234.6117
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614541
| -622284
|----
|'''Geometry'''
|[[Image:Hexadiene_anti2.jpg|400px]]
|[[Image:Hexadiene_anti2.jpg|400px]]
|----
|Bond lengths C1-C2 & C12-C13/Å
|1.33
|1.32
|----
|Bond lengths C2-C6 & C7-C12/Å
|1.50
|1.51
|----
|Bond length C6-C7/Å
|1.55
|1.55
|----
|Bond angles C1-C2-C6 & C7-C12-C13/°
|125.3
|124.8
|----
|Bond angles C2-C6-C7 & C6-C7-C12/°
|112.7
|111.4
|----
|Dihedral C1-C2-C6-C7 & C6-C7-C12-C13/°
|118.5
|114.7
|----
|Dihedral C2-C6-C7-C12/°
|180.0
|180.0
|----
|}

To view DFT optimisation file: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI2_DFT_opt.LOG
To view DFT freq file: {{DOI|10042/to-6215}}

As can be seen, the final energy calculated by DFT is lower than that by HF, and their geometry, though very similar, differs slightly e.g. dihedral angles at the sides, vinyl bond angles, and alkene bond angles.

===Thermochemistry of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

Thermochemistry data is obtained by doing a frequency calculation on the optimised structure, at the same level of theory DFT/B3LYP/6-31G*, at temperatures 0K and 298.15K respectively. Data is summarised in table 1.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.3: Comparison of Thermochemistry Data of Anti2 At 0K & 298.15K''
|----
|
|'''T=0K'''
|'''T=298.15K'''
|----
|Sum of electronic and zero point energies/Hartree
| -234.4663
| -234.4692
|----
|Sum of electronic and thermal energies/Hartree
| -234.4607
| -234.4619
|----
|Sum of electronic and thermal enthalpies/Hartree
| -234.4597
| -234.4609
|----
|Sum of electronic and thermal free energies/Hartree
| -234.4955
| -234.5008
|----
|}

To view opt+freq file at 0K: {{DOI|10042/to-6216}}

It is observed that the energies are higher for the system at 0K than at 298.15K, indicating that the molecule absorbs more heat at higher temperatures and thus thermal contribution to the energies increases.

==''Chair'' Product==

===Optimisation of Chair Product===

Two approaches are used to optimise the chair transition state structure, both at the level of theory of HF/3-21G, at default temperature of 198.15K and 1.0atm pressure. One was done using optimisation to TS(Berny) directly, calculating the force constant only once. The other was done using the redundant coordinate, first to freeze the breaking/forming bond at 2.2Å and optimise to minimum, then finally removing the freeze and optimise the whole structure to a minimum transition state TS(Berny). Results are summaried in table 1.4:

{| class="wikitable" border="1"
|+ ''Table 1.4: Comparison of Optimisation of Chair- Transition State with Different Conditions''
|----
|
|'''TS (Berny)'''<br />{{DOI|10042/to--6217}}
|'''TS (Berny) with Redundant<br />Coordinate''' <br />{{DOI|10042/to--6218}}<br />{{DOI|10042/to--6219}}
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_ts_TSBerny.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_TSBerny_(mod_red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Optimised Energy/Hartree
| -231.6193
| -231.6193
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614347.0313
| -614347.0313
|----
|Bond forming length/Å
|2.02043
|2.02069
|----
|Bond breaking length/Å
|2.02051
|2.02074
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -818
| -818
|----
|Vibration Figure
|[[Image:Chair_ts_HF_freq818.jpg|150px]]
|[[Image:Chair_ts_HF_freq818(mod-red).jpg|150px]]
|----
|}

Both sets of results yielded very similar outcomes, suggesting that either method is reliable and sufficient. Looking at the imaginary frequency at -818cm<sup>-1</sup>, it can be seen that the bond breaking and forming motions occur in a concerted manner simultaneously and asynchronously, corresponding to the definition of pericyclic sigmatropic reaction.

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

Intrinsic reaction coordinate (IRC) calculation was run to track the energy of the transition state along its reaction pathway, reaching a minimum value. The number of points along IRC was set at 100, and a minimum was reached successfully. This calculation is repeated for both force constant calculation for once and for always. See table 1.5

{| class="wikitable" border="1"
|+''Table 1.5: Comparison of IRC results for Chair-TS ''
|----
|'''Force Constant Computation'''
|'''Energy/Hartree'''
|'''Energy along IRC Path'''
|'''Gradient'''
|'''Minimum Structure'''
|----
|'''Once''' <br /> {{DOI|10042/to-6220}}
| -231.6851
|[[Image:Chair_IRC_(ONCE)_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_min_str.jpg|200px]]
|----
|'''Always'''<br /> {{DOI|10042/to-6221}}
| -231.6917
|[[Image:Chair_IRC_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_minimum_structure.jpg|200px]]
|----
|}

For the “force constant=once” calculation, even though IRC points set to 100, only 21 steps were performed by the programme and the energy graph did not plateau to a minimum, and its corresponding RMS gradient definitely did not reach zero. Hence, it did not successfully reach a minimum geometry. However, for the “force constant=always” calculation, the gradient has fallen to zero, it shows that the total energy of the structure has indeed fallen to a plateau minimum successfully, due to force constants being calculated at each and every step. This successful calculation can then be related to the gauche3 conformer obtained above, which has a comparable energy of -231.6927.

==Optimisation of ''Boat'' Product==

The boat transition state is optimised using yet another approach – with QST2 at the same level of theory. Here, it was ensured that the numbering of the atoms in reactant and that in product match exactly; otherwise, the calculation fails. Summary of optimisation is shown in table 1.6:

{| class="wikitable" border="1"
|+''Table 1.6: Optimisation results of Boat-Transition State by QST2''
|----
|Energy /Hartree
| -231.6028
|----
|Energy/kJ mol<sup>-1</sup>
| -614303.2667
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -840
|----
|Bond forming/breaking length/Å
|2.14
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Boat.mol</uploadedFileContents></jmolApplet></jmol>
|----
|D-space link
|{{DOI|10042/to-6223}}
|----
|}

Looking at the imaginary frequency at -840cm<sup>-1</sup>, the vibration motion stretches horizontally in plane in concerted and asynchronous manner, indicating again that the breaking and formation of bond is concerted and simultaneous, confirming the definition of Cope rearrangement.

==Activation Energy==

Thermochemical energies of chair/boat transition states are compared with reactant, and each corresponding activation energies for chair and boat pathways are calculated and compared, across T=0K and T=298.15K, all done at the same level of theory at DFT/B3LYP/6-31G*. The summary of energies are shown in table 1.7:

{| class="wikitable" border="1"
|+''Table 7: Comparison of Activation Energies of Chair & Boat TS at 0K & 298.15K''
|----
|
|
| colspan="4" | '''T=0K'''
| colspan="3" | '''T=298.15K'''
|----
|
| '''Electronic Energy'''
|'''Sum of electronic & zero-point energies/Hartree'''
|'''Activation Energy E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|'''E<sub>A</sub>(expt, 0K)/kcal mol<sup>-1</sup>'''
|'''Sum of electronic & thermal free energies/Hartree'''
|'''E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|----
|'''Chair TS'''
| -231.6193
| -234.4119
|0.0544
|34.11
|33.5+-0.5
| -234.4082
|0.0537
|33.67
|----
|'''Boat TS'''
| -231.6028
| -234.3990
|0.0673
|42.23
|44.7 +- 2.0
| -234.3960
|0.0659
|41.32
|----
|'''Reactant'''
| -231.6925
| -234.4663
|NA
|NA
|NA
| -234.4619
|NA
|NA
|----
|}

E<sub>A</sub> for T=0K is done by taking the difference between the sum of electronic & zero-point energies of reactant anti2 1,5-hexadiene and the chair/boat transition state, followed by converting the units from Hartree to kcal/mol (multiply by 627.509). E(A) for T=298.15K is done by the difference between their sum of electronic & thermal energies instead. The activation energy is found to be always higher for the boat than chair conformation, which agrees with expectation that chair-cyclohexane is usually the more stable form, with minimal steric clash due to the staggered H’s. In boat TS, the eclipsed hydrogens will tend to result in steric or torsional strain, causing destabilisation of the structure.

At T=0K, computational values found here are in good agreement with literature experimental values, with the values being just very slightly out of range from experimental data. This could be due to many trivial causes, especially since computational methods generate slightly different results each time it is run, or it could be due to the degree of accuracy imposed by the level of theory used. However, generally, the trend that chair-TS is more stable than its boat counterpart is well followed by computational methods here.

To view Chair-TS opt+freq by DFT:{{DOI|10042/to-6224}}{{DOI|10042/to-6225}} <br />
To view Boat-TS opt+freq by DFT: {{DOI|10042/to-6222}}{{DOI|10042/to-6226}}

=Diels Alder Reaction=

The Diels-Alder reaction is a [4+2] cycloaddition, between a conjugated diene and an dienophile, involving the concerted formation of two (or more) σ-bonds between the termini of two or more conjugated π-systems. The reaction investigated here is between cis-butadiene and ethylene; and the reaction between 1,3-cyclohexadiene and maleic anhydride is studied for its regioselectivity property.

==Ex 1: HOMO & LUMO of Reactants==

The HOMO and LUMO for reactants (cis-butadiene and ethylene) are obtained upon their optimised structure, done at the level of theory of semi-empirical/AM1. Their properties are summarised in table 2.1:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.1: HOMO & LUMO of cis-Butadiene & Ethylene (Reactants)''
|----
|
|'''Cis-Butadiene'''
|'''Ethylene'''
|----
|'''HOMO'''
|[[Image:YL_Butadiene_HOMO.jpg|150px]]
|[[Image:YL_Ethene_HOMO.jpg|150px]]
|----
|Energy /Hartree
| -0.34381
| -0.38777
|----
|Symmetry
|anti-symmetric
|symmetric
|----
|'''LUMO'''
|[[Image:YL_Butadiene_LUMO.jpg|150px]]
|[[Image:YL_Ehene_LUMO.jpg|150px]]
|----
|Energy /Hartree
|0.01708
|0.05284
|----
|Symmetry
|symmetric
|anti-symmetric
|----
|}

To view cis-butadiene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_MO.LOG <br />
To view ethylene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_MO.LOG

==Ex 2: Transition State Geometries==

===Optimisation/Geometry===

The transition state of the reaction is optimised with TS(Berny) with specified redundant coordinate method – the breaking/forming bonds are frozen to 2.1Å first and optimised to a minimum, before optimising with bond derivative to TS(Berny). The level of theory used is semiempirical/AM1. The optimisation summary shown in table 2.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.2: Optimisation of transition state''
|----
|Structure
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>TS_(mod-red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
|0.11163
|----
|Energy/kJ mol<sup>-1</sup>
|296
|----
|Bond forming length/Å
|2.1189
|----
|Bond breaking length/Å
|2.1196
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -956
|----
|Vibration
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|}

To view optimisation for TS: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:TS_OPT2_MOD-RED.LOG

The optimised geometry is further investigated in table 2.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.3: Geometry of optimised transition state''
|----
|
|'''Transition State'''
|'''Butadiene'''
|'''Ethene'''
|----
|Structure
|[[Image:TS_with_labels.jpg|350px]]
|[[Image:Butadiene_with_labels.jpg|250px]]
|[[Image:Ethene_with_labels.jpg|250px]]
|----
|Bond length C6-C4/Å
|1.4
|1.45
|NA
|----
|Bond length C1-C4/Å
|1.38
|1.34
|NA
|----
|Bond length C6-C8/Å
|1.38
|1.34
|NA
|----
|Bond length C11-C12/Å
|1.38
|NA
|1.33
|----
|Bond angle (C4-C1-C12 & C6-C8-C11)/
|99.3
|
|NA
|----
|Bond forming (C6-C11)/Å
|2.12
|NA
|NA
|----
|Bond breaking (C1-C12)/Å
|2.12
|NA
|NA
|----
|}

From literature <ref name="0031-8949">F.H. Allen, Olga Kennard, D.G. Watson, ''J. Chem. Soc. Perkin Trans. II'', 1987, S1-S18 </ref><ref name="dewarcc">M. J. S. Dewar and H. N. Schmeising, ''Tetrahedron'', 1960, '''11''', {{DOI|10.1016/0040-4020(60)89012-6}} </ref>, a typical sp<sup>3</sup> C-C bond is 1.544Å long, while sp<sup>2</sup> C-C bond is 1.338Å long, and that the van der walls radius of carbon is 1.7Å. Judging from the data obtained, the partially formed bonds (2.12Å) are all longer than typical sp<sup>3</sup> C-C bond but shorter than twice the van der waals radius of carbon. It thus suggests that there is indeed bonding/overlap, and since a bond is still in the process of forming, it is an incomplete C-C bond, and hence, longer than usual.

===Molecular Orbitals===
The HOMO and LUMO orbitals of the transition states are shown in table 2.4:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.4: HOMO and LUMO of transition state''
|----
|
|'''HOMO'''
|'''LUMO'''
|----
|MO
|[[Image:YL_TS_HOMO.jpg|150px]]
|[[Image:YL_TS_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.32392
|0.02315
|----
|Symmetry (wrt plane)
|antisymmetric
|symmetric
|----
|}

Comparing to the HOMO and LUMO orbitals of the reactants shown earlier, it can be seen that the HOMO of transition state consists of the antisymmetric HOMO of cis-butadiene and the antisymmetric LUMO of ethane, forming a π(butadiene, diene)-π*(ethylene, dienophile) interaction, typical of cycloadditions. Since 6π electrons are involved (i.e. 4n+2 where n=1), under thermal conditions it would proceed via the Huckel route with all suprafacial components, and hence the bond forms on the same face of both the butadiene and ethylene π systems.

The HOMO(butadiene)-LUMO(ethylene) interaction is more favourable than the other way round due to the smaller energy gap:
*HOMO(butadiene)-LUMO(ethene) = 0.39665 Hartree (smaller, more favourable)
*HOMO(ethene)-LUMO(butadiene)= 0.40485 Hartree (larger, not favoured)

===Vibrational Frequency===
{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.5: Vibrations of Transition State''
|----
|'''Frequency/cm<sup>-1</sup>'''
|'''Description'''
|'''Mode of Vibration'''
|----
| -956
|Synchronous and concerted bond formation
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|147
|Rocking: ethene unit rocks about butadiene unit.
|[[Image:TS_vib_freq-147.jpg|150px]]
|----
|}

From the imaginary frequency of vibration, it can be seen that both bonds form in a concerted synchronous manner, where the ethene carbons move towards or away from butadiene unit together. This shows the concerted formation of bonds, as expected of a Diels Alder cycloaddition, as defined in the introduction of this section.

===Activation Energy===
The activation energy for the transition state is calculated at 298.15K at AM1 and at a higher level of B3LYP/6-31G* levels respectively for comparison. See table 2.6:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.6: Summary of energies''
|----
|
| colspan="2" | '''AM1'''
| colspan="2" | '''B3LYP/6-31G*'''
|-
|
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies '''
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies'''
|----
|'''Reactants (Cis-Butadiene+Ethene)''' /Hartree
|0.0750
|0.2594
| -234.5991
| -234.4299
|----
|'''Transition State''' /Hartree
|0.1116
|0.2594
| -234.5734
| -234.4876
|----
|'''Activation Energy E<sub>A</sub> /kcal mol<sup>-1</sup>
| --
|22.95
| --
|16.11
|----
|}

To view opt+freq of TS by DFT:{{DOI|10042/to-6230}} <br />
To view opt+freq of cis butadiene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_FREQ-DFT.LOG <br />
To veiw opt+freq of ethene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_OPT%2BFREQ_%28DFT%29.LOG

The energy differences between the transition state and ground state reactant are the activation energy, and only these differences can be compared (but not the absolute energies). It is found that the E<sub>A</sub> found at higher level of theory is smaller. However, both are of relatively similar orders of magnitude.

==Ex 3: Regioselectivity==

In this small exercise, the Diels-Alder cycloaddition between 1,3-Cyclohexadiene and Maleic Anhydride are investigated -- pertaining to its regioselectivity of exo or endo product. From literature, both theoretical and experimental, it is well known that endo-products are preferentially formed as the fast, kinetic product due to transition state stabilisation and thus lowered activation energy.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref><ref name="woodward hoffman">R. Hoffman, R. Woodward, ''J. Chem. Soc.'', 1965, '''87''', 4388-4389: {{DOI|10.1021/ja00947a033}}</ref> This will be verified here via computational predictions.

===Optimisation===
Both the endo and exo transition states are optimised using AM1 and B3LYP/6-31G* repectively, by optimising to TS(Berny). Summary to the calculations are in table 2.7.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.7: Comparison of optimisation of transition states''
|----
|
| colspan="2" | '''Exo-TS'''
| colspan="2" | '''Endo-TS'''
|----
|
|'''AM1'''
|'''B3LYP/6-31G*'''
|'''AM1'''
|'''B3LYP/6-31G*'''
|----
| Jmol
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Exo_TS.mol</uploadedFileContents></jmolApplet></jmol>
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Endo_TS.mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
| -0.0505
| -612.6793
| -0.0516
| -612.6833
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -811
| -448
| -806
| -446
|----
|Vibrations
| colspan="2" | [[Image:YL_Exo_vib.jpg|150px]]
| colspan="2" | [[Image:YL_Endo_vib.jpg|170px]]
|----
|}

To view opt+freq of exo-TS by AM1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:EXO_TS_OPT%2BFREQ.LOG
To view opt+freq of endo-TS by AM1: (file size too big to upload here)
To view opt+freq of exo-TS by DFT:{{DOI|10042/to-6232}}
To view opt+freq of endo-TS by DFT:{{DOI|10042/to-6231}}

As can be seen from above, endo transition state is more stable (i.e. lower in energy) than exo transition state, indicating that endo transition state is favoured – a lower transition state energy leads to reduced activation energy, and a preferred kinetic product.

Imaginary frequencies of exo and endo states show that the bond formation occurs in a concerted, synchronous manner, agreeing with the definition of Diels Alder cycloaddition definition.

===Molecular Orbitals===
HOMO and LUMO orbitals of both exo and endo TS are summaried in table 2.8 below:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.8: HOMO and LUMO of Exo- & Endo- TS''
|----
|
| colspan="2" | '''Exo TS'''
| colspan="2" | '''Endo TS'''
|----
|
|'''HOMO'''
|'''LUMO'''
|'''HOMO'''
|'''LUMO'''
|----
|Structure
|[[Image:YL_Exo_HOMO.jpg|150px]]
|[[Image:YL_Exo_LUMO.jpg|150px]]
|[[Image:YL_Endo_HOMO.jpg|150px]]
|[[Image:YL_Endo_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.3427
| -0.0404
| -0.3451
| -0.0357
|----
|Symmetry wrt Plane
|Antisymmetric
|Antisymmetric
|Antisymmetric
|Antisymmetric
|----
|}

All HOMO and LUMO orbitals of exo and endo TS are antisymmetric with respect to the mirror plane. Comparing with the HOMO and LUMOs of starting materials, it is observed that the transition state HOMOs consist of the interaction between HOMO of 1,3-cyclohexadiene and LUMO of maleic anhydride. Hence, this Diels Alder cycloaddition is between the HOMO of the diene and the LUMO of the dienophile.

===Geometries===

[[Image:Exo_geometry_labelled.jpg|380px]] [[Image:Endo_geometry_labelled.jpg|360px]]

Figure on the left: exo-TS. Figure on the right: endo-TS.

Note: All lengths are labelled in Å, all angles in degrees /°. All H atoms are not shown.

Since this is a concerted reaction, the breaking of maleic anhydride/cyclohexadiene bonds and forming of new σ-bonds are expected to be about the same length – which is indeed observed in both exo and endo isomers, where the forming/breaking length occurs at about 1.40Å (1.39 to 1.41Å).

It is noted that the distance between the maleic anhydride and the cyclohexadiene component above it is 2.95Å in exo isomer but 2.89Å in the endo isomer, suggesting that the steric crowding would cause endo-isomer to be less stable. However in contrary endo-TS is in fact more stable, and this phenomenon is explained by the stabilising overlap between π* (C=O) of maleic anhydride and π(C-C) of 1,3-cyclohexadiene, otherwise known as secondary orbital overlap, as demonstrated in the past by e.g. Fox et al <ref>M.A. Fox, R.Cardona, N.J. KIwiet, ''J. Org. Chem.'' 1987, '''52''', 1469-1474.{{DOI|10.1021/jo00384a016}}</ref>. This stabilises the whole system, and hence endo isomer is of a lower energy than its exo counterpart. It is to be noted that secondary orbital overlap occurs on top of the HOMO-LUMO primary orbital interaction of the frontier orbitals. A simplified analogous diagram below, by Fox et al, shows the stabilising interactions with secondary orbital contribution in endo isomer as compared to the exo-isomer.

[[Image:Secondary_orb_exo.jpg|300px]] [[Image:Secondary_orb_endo.jpg|300px]]

Other than these, most of the bond lengths are similar in both structures. More significant differences appear in the newly-formed bond angle between the two components -- 92.7° in exo-TS and 96.7° in endo-TS.

In conclusion, this exercise proves that, in agreement with literature, endo product is favoured in a Diels Alder reaction, due to its transition state being more stabilised (i.e. lower in energy) than its exo isomer, hence endo product is the kinetic product while exo is thermodynamic product.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref>

=Conclusions=

In conclusions, computational methods using Gaussian calculations (with various levels of theory and different approaches) can be used to predict the viability, stability and feasibility of transition states for a pericyclic reaction, eg cycloaddition (Diels-Alder) or sigmatropic Cope rearrangement. Both steric and electronic aspects can be studied by investigating energies, geometries, structures, frontier orbitals, and reaction coordinate/reaction pathways, providing a more holistic understanding of a reaction, especially if it is kinetically controlled by studying its possible transition states.

=References=
<references />

File:EXO TS OPT+FREQ.LOG

2010-12-16T01:16:36Z

Ly308:

Rep:Mod:mariuspontmercy

2010-12-16T01:12:44Z

Ly308: /* Activation Energy */

'''''Note to Marker''': As most of the calculations are rather fast, they are mostly run on the laptop instead of on the college SCAN platform. As a result, most of these do not have a D-space link to the public repository. The major output files (eg .log or .chk) are hence uploaded as an "image" link and you can access it by clicking it/downloading it -- as per instructions given to us in Module 2''

=Cope Rearrangement=

Cope rearrangement is a percicyclic reaction of [3,3] sigmatropic nature, defined by its characteristic concerted migration of a group from one point of attachment to a conjugated system to another point of attachment. It involves the breaking and formation of one σ-bond each during the process. Here, the reactant, 1,5-hexadiene, is optimised to its minimum-energy (or most stable) conformation before a transition state is calculated. Intrinsic reaction coordinate (IRC) is finally used to relate the starting material/final product comformation to the transitions tate.

==Optimisation of Reactant==
===Conformers of 1,5-hexadiene: HF/3-21G===

1,5-Hexadiene was optimised with an “anti” linkage (i.e. antiperiplanar dihedral angle at its central four carbons), by using Hartree Fork (HF) and a basis set of 3-21G. The ''Anti-1'' conformer was obtained and symmetrised in Gaussview. Next, a gauche conformer (''Gauche3'') was obtained in a similar manner by altering its central dihedral angle. The results are summarized in table 1.1.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.1: Optimisation of 1,5-Hexadiene Conformers''
|----
|
|'''''Anti-1'''''
|'''''Anti-2'''''
|'''''Gauche-3'''''
|----
|Optimised Energy /Hartree
| -231.6926
| -231.6925
| -231.6927
|----
|Optimised Energy /kJ mol<sup>-1</sup>
| -614541
| -614541
| -614542
|----
|Point Group Symmetry
|C<sub>2</sub>
|C<sub>i</sub>
|C<sub>1</sub>
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Hexadiene_anti1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_anti2.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|----
|}

Counter intuitively, the ''gauche3'' conformer is more stable than the ''anti1'', even though the steric crowding in ''gauche3'' could lead to repulsions and destabilisation. Literature results, eg reported by Rocque et al <ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> have also famously proven that gauche conformers are more stable and preferred than anti ones. This is explained by the stereoelectronic effect, which involves stabilisating interaction between the π-electrons of the C=C and vinyl protons, otherwise known as CH-π interaction, thus stabilising ''gauche3'' instead.<ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> This is confirmed by performing another optimisation on an anti-conformer – ''anti2''. And indeed, it is observed that ''anti2'' and ''anti1'' are comparable in their energies, and ''gauche3'' is lower in energy.

To view optimisation files for anti1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI1.LOG

To view optimisation files for anti2: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI.2LOG

To view optimisation files for gauche3: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_GAUCHE3_OPT.LOG

===Optimisation of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

''Anti2'' conformer is optimised by B3LYP method, with basis set of 6-31G*. A summary and comparison of both methods (HF/3-21G and DFT/B3LYP/6-31G*) are shown in table 1.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.2: Comparison of optimisation properties of anti2 conformer, by different methods''
|----
|'''Method'''
|'''HF/3-21G'''
|'''DFT/B3LYP/6-31G*'''
|----
|Optimised Energy/Hartree
| -231.6925
| -234.6117
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614541
| -622284
|----
|'''Geometry'''
|[[Image:Hexadiene_anti2.jpg|400px]]
|[[Image:Hexadiene_anti2.jpg|400px]]
|----
|Bond lengths C1-C2 & C12-C13/Å
|1.33
|1.32
|----
|Bond lengths C2-C6 & C7-C12/Å
|1.50
|1.51
|----
|Bond length C6-C7/Å
|1.55
|1.55
|----
|Bond angles C1-C2-C6 & C7-C12-C13/°
|125.3
|124.8
|----
|Bond angles C2-C6-C7 & C6-C7-C12/°
|112.7
|111.4
|----
|Dihedral C1-C2-C6-C7 & C6-C7-C12-C13/°
|118.5
|114.7
|----
|Dihedral C2-C6-C7-C12/°
|180.0
|180.0
|----
|}

To view DFT optimisation file: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI2_DFT_opt.LOG
To view DFT freq file: {{DOI|10042/to-6215}}

As can be seen, the final energy calculated by DFT is lower than that by HF, and their geometry, though very similar, differs slightly e.g. dihedral angles at the sides, vinyl bond angles, and alkene bond angles.

===Thermochemistry of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

Thermochemistry data is obtained by doing a frequency calculation on the optimised structure, at the same level of theory DFT/B3LYP/6-31G*, at temperatures 0K and 298.15K respectively. Data is summarised in table 1.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.3: Comparison of Thermochemistry Data of Anti2 At 0K & 298.15K''
|----
|
|'''T=0K'''
|'''T=298.15K'''
|----
|Sum of electronic and zero point energies/Hartree
| -234.4663
| -234.4692
|----
|Sum of electronic and thermal energies/Hartree
| -234.4607
| -234.4619
|----
|Sum of electronic and thermal enthalpies/Hartree
| -234.4597
| -234.4609
|----
|Sum of electronic and thermal free energies/Hartree
| -234.4955
| -234.5008
|----
|}

To view opt+freq file at 0K: {{DOI|10042/to-6216}}

It is observed that the energies are higher for the system at 0K than at 298.15K, indicating that the molecule absorbs more heat at higher temperatures and thus thermal contribution to the energies increases.

==''Chair'' Product==

===Optimisation of Chair Product===

Two approaches are used to optimise the chair transition state structure, both at the level of theory of HF/3-21G, at default temperature of 198.15K and 1.0atm pressure. One was done using optimisation to TS(Berny) directly, calculating the force constant only once. The other was done using the redundant coordinate, first to freeze the breaking/forming bond at 2.2Å and optimise to minimum, then finally removing the freeze and optimise the whole structure to a minimum transition state TS(Berny). Results are summaried in table 1.4:

{| class="wikitable" border="1"
|+ ''Table 1.4: Comparison of Optimisation of Chair- Transition State with Different Conditions''
|----
|
|'''TS (Berny)'''<br />{{DOI|10042/to--6217}}
|'''TS (Berny) with Redundant<br />Coordinate''' <br />{{DOI|10042/to--6218}}<br />{{DOI|10042/to--6219}}
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_ts_TSBerny.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_TSBerny_(mod_red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Optimised Energy/Hartree
| -231.6193
| -231.6193
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614347.0313
| -614347.0313
|----
|Bond forming length/Å
|2.02043
|2.02069
|----
|Bond breaking length/Å
|2.02051
|2.02074
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -818
| -818
|----
|Vibration Figure
|[[Image:Chair_ts_HF_freq818.jpg|150px]]
|[[Image:Chair_ts_HF_freq818(mod-red).jpg|150px]]
|----
|}

Both sets of results yielded very similar outcomes, suggesting that either method is reliable and sufficient. Looking at the imaginary frequency at -818cm<sup>-1</sup>, it can be seen that the bond breaking and forming motions occur in a concerted manner simultaneously and asynchronously, corresponding to the definition of pericyclic sigmatropic reaction.

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

Intrinsic reaction coordinate (IRC) calculation was run to track the energy of the transition state along its reaction pathway, reaching a minimum value. The number of points along IRC was set at 100, and a minimum was reached successfully. This calculation is repeated for both force constant calculation for once and for always. See table 1.5

{| class="wikitable" border="1"
|+''Table 1.5: Comparison of IRC results for Chair-TS ''
|----
|'''Force Constant Computation'''
|'''Energy/Hartree'''
|'''Energy along IRC Path'''
|'''Gradient'''
|'''Minimum Structure'''
|----
|'''Once''' <br /> {{DOI|10042/to-6220}}
| -231.6851
|[[Image:Chair_IRC_(ONCE)_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_min_str.jpg|200px]]
|----
|'''Always'''<br /> {{DOI|10042/to-6221}}
| -231.6917
|[[Image:Chair_IRC_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_minimum_structure.jpg|200px]]
|----
|}

For the “force constant=once” calculation, even though IRC points set to 100, only 21 steps were performed by the programme and the energy graph did not plateau to a minimum, and its corresponding RMS gradient definitely did not reach zero. Hence, it did not successfully reach a minimum geometry. However, for the “force constant=always” calculation, the gradient has fallen to zero, it shows that the total energy of the structure has indeed fallen to a plateau minimum successfully, due to force constants being calculated at each and every step. This successful calculation can then be related to the gauche3 conformer obtained above, which has a comparable energy of -231.6927.

==Optimisation of ''Boat'' Product==

The boat transition state is optimised using yet another approach – with QST2 at the same level of theory. Here, it was ensured that the numbering of the atoms in reactant and that in product match exactly; otherwise, the calculation fails. Summary of optimisation is shown in table 1.6:

{| class="wikitable" border="1"
|+''Table 1.6: Optimisation results of Boat-Transition State by QST2''
|----
|Energy /Hartree
| -231.6028
|----
|Energy/kJ mol<sup>-1</sup>
| -614303.2667
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -840
|----
|Bond forming/breaking length/Å
|2.14
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Boat.mol</uploadedFileContents></jmolApplet></jmol>
|----
|D-space link
|{{DOI|10042/to-6223}}
|----
|}

Looking at the imaginary frequency at -840cm<sup>-1</sup>, the vibration motion stretches horizontally in plane in concerted and asynchronous manner, indicating again that the breaking and formation of bond is concerted and simultaneous, confirming the definition of Cope rearrangement.

==Activation Energy==

Thermochemical energies of chair/boat transition states are compared with reactant, and each corresponding activation energies for chair and boat pathways are calculated and compared, across T=0K and T=298.15K, all done at the same level of theory at DFT/B3LYP/6-31G*. The summary of energies are shown in table 1.7:

{| class="wikitable" border="1"
|+''Table 7: Comparison of Activation Energies of Chair & Boat TS at 0K & 298.15K''
|----
|
|
| colspan="4" | '''T=0K'''
| colspan="3" | '''T=298.15K'''
|----
|
| '''Electronic Energy'''
|'''Sum of electronic & zero-point energies/Hartree'''
|'''Activation Energy E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|'''E<sub>A</sub>(expt, 0K)/kcal mol<sup>-1</sup>'''
|'''Sum of electronic & thermal free energies/Hartree'''
|'''E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|----
|'''Chair TS'''
| -231.6193
| -234.4119
|0.0544
|34.11
|33.5+-0.5
| -234.4082
|0.0537
|33.67
|----
|'''Boat TS'''
| -231.6028
| -234.3990
|0.0673
|42.23
|44.7 +- 2.0
| -234.3960
|0.0659
|41.32
|----
|'''Reactant'''
| -231.6925
| -234.4663
|NA
|NA
|NA
| -234.4619
|NA
|NA
|----
|}

E<sub>A</sub> for T=0K is done by taking the difference between the sum of electronic & zero-point energies of reactant anti2 1,5-hexadiene and the chair/boat transition state, followed by converting the units from Hartree to kcal/mol (multiply by 627.509). E(A) for T=298.15K is done by the difference between their sum of electronic & thermal energies instead. The activation energy is found to be always higher for the boat than chair conformation, which agrees with expectation that chair-cyclohexane is usually the more stable form, with minimal steric clash due to the staggered H’s. In boat TS, the eclipsed hydrogens will tend to result in steric or torsional strain, causing destabilisation of the structure.

At T=0K, computational values found here are in good agreement with literature experimental values, with the values being just very slightly out of range from experimental data. This could be due to many trivial causes, especially since computational methods generate slightly different results each time it is run, or it could be due to the degree of accuracy imposed by the level of theory used. However, generally, the trend that chair-TS is more stable than its boat counterpart is well followed by computational methods here.

To view Chair-TS opt+freq by DFT:{{DOI|10042/to-6224}}{{DOI|10042/to-6225}} <br />
To view Boat-TS opt+freq by DFT: {{DOI|10042/to-6222}}{{DOI|10042/to-6226}}

=Diels Alder Reaction=

The Diels-Alder reaction is a [4+2] cycloaddition, between a conjugated diene and an dienophile, involving the concerted formation of two (or more) σ-bonds between the termini of two or more conjugated π-systems. The reaction investigated here is between cis-butadiene and ethylene; and the reaction between 1,3-cyclohexadiene and maleic anhydride is studied for its regioselectivity property.

==Ex 1: HOMO & LUMO of Reactants==

The HOMO and LUMO for reactants (cis-butadiene and ethylene) are obtained upon their optimised structure, done at the level of theory of semi-empirical/AM1. Their properties are summarised in table 2.1:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.1: HOMO & LUMO of cis-Butadiene & Ethylene (Reactants)''
|----
|
|'''Cis-Butadiene'''
|'''Ethylene'''
|----
|'''HOMO'''
|[[Image:YL_Butadiene_HOMO.jpg|150px]]
|[[Image:YL_Ethene_HOMO.jpg|150px]]
|----
|Energy /Hartree
| -0.34381
| -0.38777
|----
|Symmetry
|anti-symmetric
|symmetric
|----
|'''LUMO'''
|[[Image:YL_Butadiene_LUMO.jpg|150px]]
|[[Image:YL_Ehene_LUMO.jpg|150px]]
|----
|Energy /Hartree
|0.01708
|0.05284
|----
|Symmetry
|symmetric
|anti-symmetric
|----
|}

To view cis-butadiene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_MO.LOG <br />
To view ethylene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_MO.LOG

==Ex 2: Transition State Geometries==

===Optimisation/Geometry===

The transition state of the reaction is optimised with TS(Berny) with specified redundant coordinate method – the breaking/forming bonds are frozen to 2.1Å first and optimised to a minimum, before optimising with bond derivative to TS(Berny). The level of theory used is semiempirical/AM1. The optimisation summary shown in table 2.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.2: Optimisation of transition state''
|----
|Structure
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>TS_(mod-red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
|0.11163
|----
|Energy/kJ mol<sup>-1</sup>
|296
|----
|Bond forming length/Å
|2.1189
|----
|Bond breaking length/Å
|2.1196
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -956
|----
|Vibration
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|}

To view optimisation for TS: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:TS_OPT2_MOD-RED.LOG

The optimised geometry is further investigated in table 2.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.3: Geometry of optimised transition state''
|----
|
|'''Transition State'''
|'''Butadiene'''
|'''Ethene'''
|----
|Structure
|[[Image:TS_with_labels.jpg|350px]]
|[[Image:Butadiene_with_labels.jpg|250px]]
|[[Image:Ethene_with_labels.jpg|250px]]
|----
|Bond length C6-C4/Å
|1.4
|1.45
|NA
|----
|Bond length C1-C4/Å
|1.38
|1.34
|NA
|----
|Bond length C6-C8/Å
|1.38
|1.34
|NA
|----
|Bond length C11-C12/Å
|1.38
|NA
|1.33
|----
|Bond angle (C4-C1-C12 & C6-C8-C11)/
|99.3
|
|NA
|----
|Bond forming (C6-C11)/Å
|2.12
|NA
|NA
|----
|Bond breaking (C1-C12)/Å
|2.12
|NA
|NA
|----
|}

From literature <ref name="0031-8949">F.H. Allen, Olga Kennard, D.G. Watson, ''J. Chem. Soc. Perkin Trans. II'', 1987, S1-S18 </ref><ref name="dewarcc">M. J. S. Dewar and H. N. Schmeising, ''Tetrahedron'', 1960, '''11''', {{DOI|10.1016/0040-4020(60)89012-6}} </ref>, a typical sp<sup>3</sup> C-C bond is 1.544Å long, while sp<sup>2</sup> C-C bond is 1.338Å long, and that the van der walls radius of carbon is 1.7Å. Judging from the data obtained, the partially formed bonds (2.12Å) are all longer than typical sp<sup>3</sup> C-C bond but shorter than twice the van der waals radius of carbon. It thus suggests that there is indeed bonding/overlap, and since a bond is still in the process of forming, it is an incomplete C-C bond, and hence, longer than usual.

===Molecular Orbitals===
The HOMO and LUMO orbitals of the transition states are shown in table 2.4:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.4: HOMO and LUMO of transition state''
|----
|
|'''HOMO'''
|'''LUMO'''
|----
|MO
|[[Image:YL_TS_HOMO.jpg|150px]]
|[[Image:YL_TS_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.32392
|0.02315
|----
|Symmetry (wrt plane)
|antisymmetric
|symmetric
|----
|}

Comparing to the HOMO and LUMO orbitals of the reactants shown earlier, it can be seen that the HOMO of transition state consists of the antisymmetric HOMO of cis-butadiene and the antisymmetric LUMO of ethane, forming a π(butadiene, diene)-π*(ethylene, dienophile) interaction, typical of cycloadditions. Since 6π electrons are involved (i.e. 4n+2 where n=1), under thermal conditions it would proceed via the Huckel route with all suprafacial components, and hence the bond forms on the same face of both the butadiene and ethylene π systems.

The HOMO(butadiene)-LUMO(ethylene) interaction is more favourable than the other way round due to the smaller energy gap:
*HOMO(butadiene)-LUMO(ethene) = 0.39665 Hartree (smaller, more favourable)
*HOMO(ethene)-LUMO(butadiene)= 0.40485 Hartree (larger, not favoured)

===Vibrational Frequency===
{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.5: Vibrations of Transition State''
|----
|'''Frequency/cm<sup>-1</sup>'''
|'''Description'''
|'''Mode of Vibration'''
|----
| -956
|Synchronous and concerted bond formation
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|147
|Rocking: ethene unit rocks about butadiene unit.
|[[Image:TS_vib_freq-147.jpg|150px]]
|----
|}

From the imaginary frequency of vibration, it can be seen that both bonds form in a concerted synchronous manner, where the ethene carbons move towards or away from butadiene unit together. This shows the concerted formation of bonds, as expected of a Diels Alder cycloaddition, as defined in the introduction of this section.

===Activation Energy===
The activation energy for the transition state is calculated at 298.15K at AM1 and at a higher level of B3LYP/6-31G* levels respectively for comparison. See table 2.6:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.6: Summary of energies''
|----
|
| colspan="2" | '''AM1'''
| colspan="2" | '''B3LYP/6-31G*'''
|-
|
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies '''
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies'''
|----
|'''Reactants (Cis-Butadiene+Ethene)''' /Hartree
|0.0750
|0.2594
| -234.5991
| -234.4299
|----
|'''Transition State''' /Hartree
|0.1116
|0.2594
| -234.5734
| -234.4876
|----
|'''Activation Energy E<sub>A</sub> /kcal mol<sup>-1</sup>
| --
|22.95
| --
|16.11
|----
|}

To view opt+freq of TS by DFT:{{DOI|10042/to-6230}} <br />
To view opt+freq of cis butadiene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_FREQ-DFT.LOG <br />
To veiw opt+freq of ethene by DFT: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_OPT%2BFREQ_%28DFT%29.LOG

The energy differences between the transition state and ground state reactant are the activation energy, and only these differences can be compared (but not the absolute energies). It is found that the E<sub>A</sub> found at higher level of theory is smaller. However, both are of relatively similar orders of magnitude.

==Ex 3: Regioselectivity==

In this small exercise, the Diels-Alder cycloaddition between 1,3-Cyclohexadiene and Maleic Anhydride are investigated -- pertaining to its regioselectivity of exo or endo product. From literature, both theoretical and experimental, it is well known that endo-products are preferentially formed as the fast, kinetic product due to transition state stabilisation and thus lowered activation energy.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref><ref name="woodward hoffman">R. Hoffman, R. Woodward, ''J. Chem. Soc.'', 1965, '''87''', 4388-4389: {{DOI|10.1021/ja00947a033}}</ref> This will be verified here via computational predictions.

===Optimisation===
Both the endo and exo transition states are optimised using AM1 and B3LYP/6-31G* repectively, by optimising to TS(Berny). Summary to the calculations are in table 2.7.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.7: Comparison of optimisation of transition states''
|----
|
| colspan="2" | '''Exo-TS'''
| colspan="2" | '''Endo-TS'''
|----
|
|'''AM1'''
|'''B3LYP/6-31G*'''
|'''AM1'''
|'''B3LYP/6-31G*'''
|----
| Jmol
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Exo_TS.mol</uploadedFileContents></jmolApplet></jmol>
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Endo_TS.mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
| -0.0505
| -612.6793
| -0.0516
| -612.6833
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -811
| -448
| -806
| -446
|----
|Vibrations
| colspan="2" | [[Image:YL_Exo_vib.jpg|150px]]
| colspan="2" | [[Image:YL_Endo_vib.jpg|170px]]
|----
|}

As can be seen from above, endo transition state is more stable (i.e. lower in energy) than exo transition state, indicating that endo transition state is favoured – a lower transition state energy leads to reduced activation energy, and a preferred kinetic product.

Imaginary frequencies of exo and endo states show that the bond formation occurs in a concerted, synchronous manner, agreeing with the definition of Diels Alder cycloaddition definition.

===Molecular Orbitals===
HOMO and LUMO orbitals of both exo and endo TS are summaried in table 2.8 below:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.8: HOMO and LUMO of Exo- & Endo- TS''
|----
|
| colspan="2" | '''Exo TS'''
| colspan="2" | '''Endo TS'''
|----
|
|'''HOMO'''
|'''LUMO'''
|'''HOMO'''
|'''LUMO'''
|----
|Structure
|[[Image:YL_Exo_HOMO.jpg|150px]]
|[[Image:YL_Exo_LUMO.jpg|150px]]
|[[Image:YL_Endo_HOMO.jpg|150px]]
|[[Image:YL_Endo_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.3427
| -0.0404
| -0.3451
| -0.0357
|----
|Symmetry wrt Plane
|Antisymmetric
|Antisymmetric
|Antisymmetric
|Antisymmetric
|----
|}

All HOMO and LUMO orbitals of exo and endo TS are antisymmetric with respect to the mirror plane. Comparing with the HOMO and LUMOs of starting materials, it is observed that the transition state HOMOs consist of the interaction between HOMO of 1,3-cyclohexadiene and LUMO of maleic anhydride. Hence, this Diels Alder cycloaddition is between the HOMO of the diene and the LUMO of the dienophile.

===Geometries===

[[Image:Exo_geometry_labelled.jpg|380px]] [[Image:Endo_geometry_labelled.jpg|360px]]

Figure on the left: exo-TS. Figure on the right: endo-TS.

Note: All lengths are labelled in Å, all angles in degrees /°. All H atoms are not shown.

Since this is a concerted reaction, the breaking of maleic anhydride/cyclohexadiene bonds and forming of new σ-bonds are expected to be about the same length – which is indeed observed in both exo and endo isomers, where the forming/breaking length occurs at about 1.40Å (1.39 to 1.41Å).

It is noted that the distance between the maleic anhydride and the cyclohexadiene component above it is 2.95Å in exo isomer but 2.89Å in the endo isomer, suggesting that the steric crowding would cause endo-isomer to be less stable. However in contrary endo-TS is in fact more stable, and this phenomenon is explained by the stabilising overlap between π* (C=O) of maleic anhydride and π(C-C) of 1,3-cyclohexadiene, otherwise known as secondary orbital overlap, as demonstrated in the past by e.g. Fox et al <ref>M.A. Fox, R.Cardona, N.J. KIwiet, ''J. Org. Chem.'' 1987, '''52''', 1469-1474.{{DOI|10.1021/jo00384a016}}</ref>. This stabilises the whole system, and hence endo isomer is of a lower energy than its exo counterpart. It is to be noted that secondary orbital overlap occurs on top of the HOMO-LUMO primary orbital interaction of the frontier orbitals. A simplified analogous diagram below, by Fox et al, shows the stabilising interactions with secondary orbital contribution in endo isomer as compared to the exo-isomer.

[[Image:Secondary_orb_exo.jpg|300px]] [[Image:Secondary_orb_endo.jpg|300px]]

Other than these, most of the bond lengths are similar in both structures. More significant differences appear in the newly-formed bond angle between the two components -- 92.7° in exo-TS and 96.7° in endo-TS.

In conclusion, this exercise proves that, in agreement with literature, endo product is favoured in a Diels Alder reaction, due to its transition state being more stabilised (i.e. lower in energy) than its exo isomer, hence endo product is the kinetic product while exo is thermodynamic product.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref>

=Conclusions=

In conclusions, computational methods using Gaussian calculations (with various levels of theory and different approaches) can be used to predict the viability, stability and feasibility of transition states for a pericyclic reaction, eg cycloaddition (Diels-Alder) or sigmatropic Cope rearrangement. Both steric and electronic aspects can be studied by investigating energies, geometries, structures, frontier orbitals, and reaction coordinate/reaction pathways, providing a more holistic understanding of a reaction, especially if it is kinetically controlled by studying its possible transition states.

=References=
<references />

Rep:Mod:mariuspontmercy

2010-12-16T01:12:09Z

File:TS OPT2 MOD-RED.LOG

2010-12-16T01:06:01Z

Ly308:

Rep:Mod:mariuspontmercy

2010-12-16T01:04:27Z

Ly308:

'''''Note to Marker''': As most of the calculations are rather fast, they are mostly run on the laptop instead of on the college SCAN platform. As a result, most of these do not have a D-space link to the public repository. The major output files (eg .log or .chk) are hence uploaded as an "image" link and you can access it by clicking it/downloading it -- as per instructions given to us in Module 2''

=Cope Rearrangement=

Cope rearrangement is a percicyclic reaction of [3,3] sigmatropic nature, defined by its characteristic concerted migration of a group from one point of attachment to a conjugated system to another point of attachment. It involves the breaking and formation of one σ-bond each during the process. Here, the reactant, 1,5-hexadiene, is optimised to its minimum-energy (or most stable) conformation before a transition state is calculated. Intrinsic reaction coordinate (IRC) is finally used to relate the starting material/final product comformation to the transitions tate.

==Optimisation of Reactant==
===Conformers of 1,5-hexadiene: HF/3-21G===

1,5-Hexadiene was optimised with an “anti” linkage (i.e. antiperiplanar dihedral angle at its central four carbons), by using Hartree Fork (HF) and a basis set of 3-21G. The ''Anti-1'' conformer was obtained and symmetrised in Gaussview. Next, a gauche conformer (''Gauche3'') was obtained in a similar manner by altering its central dihedral angle. The results are summarized in table 1.1.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.1: Optimisation of 1,5-Hexadiene Conformers''
|----
|
|'''''Anti-1'''''
|'''''Anti-2'''''
|'''''Gauche-3'''''
|----
|Optimised Energy /Hartree
| -231.6926
| -231.6925
| -231.6927
|----
|Optimised Energy /kJ mol<sup>-1</sup>
| -614541
| -614541
| -614542
|----
|Point Group Symmetry
|C<sub>2</sub>
|C<sub>i</sub>
|C<sub>1</sub>
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Hexadiene_anti1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_anti2.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|----
|}

Counter intuitively, the ''gauche3'' conformer is more stable than the ''anti1'', even though the steric crowding in ''gauche3'' could lead to repulsions and destabilisation. Literature results, eg reported by Rocque et al <ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> have also famously proven that gauche conformers are more stable and preferred than anti ones. This is explained by the stereoelectronic effect, which involves stabilisating interaction between the π-electrons of the C=C and vinyl protons, otherwise known as CH-π interaction, thus stabilising ''gauche3'' instead.<ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> This is confirmed by performing another optimisation on an anti-conformer – ''anti2''. And indeed, it is observed that ''anti2'' and ''anti1'' are comparable in their energies, and ''gauche3'' is lower in energy.

To view optimisation files for anti1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI1.LOG

To view optimisation files for anti2: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI.2LOG

To view optimisation files for gauche3: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_GAUCHE3_OPT.LOG

===Optimisation of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

''Anti2'' conformer is optimised by B3LYP method, with basis set of 6-31G*. A summary and comparison of both methods (HF/3-21G and DFT/B3LYP/6-31G*) are shown in table 1.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.2: Comparison of optimisation properties of anti2 conformer, by different methods''
|----
|'''Method'''
|'''HF/3-21G'''
|'''DFT/B3LYP/6-31G*'''
|----
|Optimised Energy/Hartree
| -231.6925
| -234.6117
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614541
| -622284
|----
|'''Geometry'''
|[[Image:Hexadiene_anti2.jpg|400px]]
|[[Image:Hexadiene_anti2.jpg|400px]]
|----
|Bond lengths C1-C2 & C12-C13/Å
|1.33
|1.32
|----
|Bond lengths C2-C6 & C7-C12/Å
|1.50
|1.51
|----
|Bond length C6-C7/Å
|1.55
|1.55
|----
|Bond angles C1-C2-C6 & C7-C12-C13/°
|125.3
|124.8
|----
|Bond angles C2-C6-C7 & C6-C7-C12/°
|112.7
|111.4
|----
|Dihedral C1-C2-C6-C7 & C6-C7-C12-C13/°
|118.5
|114.7
|----
|Dihedral C2-C6-C7-C12/°
|180.0
|180.0
|----
|}

To view DFT optimisation file: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI2_DFT_opt.LOG
To view DFT freq file: {{DOI|10042/to-6215}}

As can be seen, the final energy calculated by DFT is lower than that by HF, and their geometry, though very similar, differs slightly e.g. dihedral angles at the sides, vinyl bond angles, and alkene bond angles.

===Thermochemistry of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

Thermochemistry data is obtained by doing a frequency calculation on the optimised structure, at the same level of theory DFT/B3LYP/6-31G*, at temperatures 0K and 298.15K respectively. Data is summarised in table 1.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.3: Comparison of Thermochemistry Data of Anti2 At 0K & 298.15K''
|----
|
|'''T=0K'''
|'''T=298.15K'''
|----
|Sum of electronic and zero point energies/Hartree
| -234.4663
| -234.4692
|----
|Sum of electronic and thermal energies/Hartree
| -234.4607
| -234.4619
|----
|Sum of electronic and thermal enthalpies/Hartree
| -234.4597
| -234.4609
|----
|Sum of electronic and thermal free energies/Hartree
| -234.4955
| -234.5008
|----
|}

To view opt+freq file at 0K: {{DOI|10042/to-6216}}

It is observed that the energies are higher for the system at 0K than at 298.15K, indicating that the molecule absorbs more heat at higher temperatures and thus thermal contribution to the energies increases.

==''Chair'' Product==

===Optimisation of Chair Product===

Two approaches are used to optimise the chair transition state structure, both at the level of theory of HF/3-21G, at default temperature of 198.15K and 1.0atm pressure. One was done using optimisation to TS(Berny) directly, calculating the force constant only once. The other was done using the redundant coordinate, first to freeze the breaking/forming bond at 2.2Å and optimise to minimum, then finally removing the freeze and optimise the whole structure to a minimum transition state TS(Berny). Results are summaried in table 1.4:

{| class="wikitable" border="1"
|+ ''Table 1.4: Comparison of Optimisation of Chair- Transition State with Different Conditions''
|----
|
|'''TS (Berny)'''<br />{{DOI|10042/to--6217}}
|'''TS (Berny) with Redundant<br />Coordinate''' <br />{{DOI|10042/to--6218}}<br />{{DOI|10042/to--6219}}
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_ts_TSBerny.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_TSBerny_(mod_red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Optimised Energy/Hartree
| -231.6193
| -231.6193
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614347.0313
| -614347.0313
|----
|Bond forming length/Å
|2.02043
|2.02069
|----
|Bond breaking length/Å
|2.02051
|2.02074
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -818
| -818
|----
|Vibration Figure
|[[Image:Chair_ts_HF_freq818.jpg|150px]]
|[[Image:Chair_ts_HF_freq818(mod-red).jpg|150px]]
|----
|}

Both sets of results yielded very similar outcomes, suggesting that either method is reliable and sufficient. Looking at the imaginary frequency at -818cm<sup>-1</sup>, it can be seen that the bond breaking and forming motions occur in a concerted manner simultaneously and asynchronously, corresponding to the definition of pericyclic sigmatropic reaction.

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

Intrinsic reaction coordinate (IRC) calculation was run to track the energy of the transition state along its reaction pathway, reaching a minimum value. The number of points along IRC was set at 100, and a minimum was reached successfully. This calculation is repeated for both force constant calculation for once and for always. See table 1.5

{| class="wikitable" border="1"
|+''Table 1.5: Comparison of IRC results for Chair-TS ''
|----
|'''Force Constant Computation'''
|'''Energy/Hartree'''
|'''Energy along IRC Path'''
|'''Gradient'''
|'''Minimum Structure'''
|----
|'''Once''' <br /> {{DOI|10042/to-6220}}
| -231.6851
|[[Image:Chair_IRC_(ONCE)_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_min_str.jpg|200px]]
|----
|'''Always'''<br /> {{DOI|10042/to-6221}}
| -231.6917
|[[Image:Chair_IRC_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_minimum_structure.jpg|200px]]
|----
|}

For the “force constant=once” calculation, even though IRC points set to 100, only 21 steps were performed by the programme and the energy graph did not plateau to a minimum, and its corresponding RMS gradient definitely did not reach zero. Hence, it did not successfully reach a minimum geometry. However, for the “force constant=always” calculation, the gradient has fallen to zero, it shows that the total energy of the structure has indeed fallen to a plateau minimum successfully, due to force constants being calculated at each and every step. This successful calculation can then be related to the gauche3 conformer obtained above, which has a comparable energy of -231.6927.

==Optimisation of ''Boat'' Product==

The boat transition state is optimised using yet another approach – with QST2 at the same level of theory. Here, it was ensured that the numbering of the atoms in reactant and that in product match exactly; otherwise, the calculation fails. Summary of optimisation is shown in table 1.6:

{| class="wikitable" border="1"
|+''Table 1.6: Optimisation results of Boat-Transition State by QST2''
|----
|Energy /Hartree
| -231.6028
|----
|Energy/kJ mol<sup>-1</sup>
| -614303.2667
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -840
|----
|Bond forming/breaking length/Å
|2.14
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Boat.mol</uploadedFileContents></jmolApplet></jmol>
|----
|D-space link
|{{DOI|10042/to-6223}}
|----
|}

Looking at the imaginary frequency at -840cm<sup>-1</sup>, the vibration motion stretches horizontally in plane in concerted and asynchronous manner, indicating again that the breaking and formation of bond is concerted and simultaneous, confirming the definition of Cope rearrangement.

==Activation Energy==

Thermochemical energies of chair/boat transition states are compared with reactant, and each corresponding activation energies for chair and boat pathways are calculated and compared, across T=0K and T=298.15K, all done at the same level of theory at DFT/B3LYP/6-31G*. The summary of energies are shown in table 1.7:

{| class="wikitable" border="1"
|+''Table 7: Comparison of Activation Energies of Chair & Boat TS at 0K & 298.15K''
|----
|
|
| colspan="4" | '''T=0K'''
| colspan="3" | '''T=298.15K'''
|----
|
| '''Electronic Energy'''
|'''Sum of electronic & zero-point energies/Hartree'''
|'''Activation Energy E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|'''E<sub>A</sub>(expt, 0K)/kcal mol<sup>-1</sup>'''
|'''Sum of electronic & thermal free energies/Hartree'''
|'''E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|----
|'''Chair TS'''
| -231.6193
| -234.4119
|0.0544
|34.11
|33.5+-0.5
| -234.4082
|0.0537
|33.67
|----
|'''Boat TS'''
| -231.6028
| -234.3990
|0.0673
|42.23
|44.7 +- 2.0
| -234.3960
|0.0659
|41.32
|----
|'''Reactant'''
| -231.6925
| -234.4663
|NA
|NA
|NA
| -234.4619
|NA
|NA
|----
|}

E<sub>A</sub> for T=0K is done by taking the difference between the sum of electronic & zero-point energies of reactant anti2 1,5-hexadiene and the chair/boat transition state, followed by converting the units from Hartree to kcal/mol (multiply by 627.509). E(A) for T=298.15K is done by the difference between their sum of electronic & thermal energies instead. The activation energy is found to be always higher for the boat than chair conformation, which agrees with expectation that chair-cyclohexane is usually the more stable form, with minimal steric clash due to the staggered H’s. In boat TS, the eclipsed hydrogens will tend to result in steric or torsional strain, causing destabilisation of the structure.

At T=0K, computational values found here are in good agreement with literature experimental values, with the values being just very slightly out of range from experimental data. This could be due to many trivial causes, especially since computational methods generate slightly different results each time it is run, or it could be due to the degree of accuracy imposed by the level of theory used. However, generally, the trend that chair-TS is more stable than its boat counterpart is well followed by computational methods here.

To view Chair-TS opt+freq by DFT:{{DOI|10042/to-6224}}{{DOI|10042/to-6225}} <br />
To view Boat-TS opt+freq by DFT: {{DOI|10042/to-6222}}{{DOI|10042/to-6226}}

=Diels Alder Reaction=

The Diels-Alder reaction is a [4+2] cycloaddition, between a conjugated diene and an dienophile, involving the concerted formation of two (or more) σ-bonds between the termini of two or more conjugated π-systems. The reaction investigated here is between cis-butadiene and ethylene; and the reaction between 1,3-cyclohexadiene and maleic anhydride is studied for its regioselectivity property.

==Ex 1: HOMO & LUMO of Reactants==

The HOMO and LUMO for reactants (cis-butadiene and ethylene) are obtained upon their optimised structure, done at the level of theory of semi-empirical/AM1. Their properties are summarised in table 2.1:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.1: HOMO & LUMO of cis-Butadiene & Ethylene (Reactants)''
|----
|
|'''Cis-Butadiene'''
|'''Ethylene'''
|----
|'''HOMO'''
|[[Image:YL_Butadiene_HOMO.jpg|150px]]
|[[Image:YL_Ethene_HOMO.jpg|150px]]
|----
|Energy /Hartree
| -0.34381
| -0.38777
|----
|Symmetry
|anti-symmetric
|symmetric
|----
|'''LUMO'''
|[[Image:YL_Butadiene_LUMO.jpg|150px]]
|[[Image:YL_Ehene_LUMO.jpg|150px]]
|----
|Energy /Hartree
|0.01708
|0.05284
|----
|Symmetry
|symmetric
|anti-symmetric
|----
|}

To view cis-butadiene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_MO.LOG <br />
To view ethylene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_MO.LOG

==Ex 2: Transition State Geometries==

===Optimisation/Geometry===

The transition state of the reaction is optimised with TS(Berny) with specified redundant coordinate method – the breaking/forming bonds are frozen to 2.1Å first and optimised to a minimum, before optimising with bond derivative to TS(Berny). The level of theory used is semiempirical/AM1. The optimisation summary shown in table 2.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.2: Optimisation of transition state''
|----
|Structure
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>TS_(mod-red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
|0.11163
|----
|Energy/kJ mol<sup>-1</sup>
|296
|----
|Bond forming length/Å
|2.1189
|----
|Bond breaking length/Å
|2.1196
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -956
|----
|Vibration
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|}

The optimised geometry is further investigated in table 2.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.3: Geometry of optimised transition state''
|----
|
|'''Transition State'''
|'''Butadiene'''
|'''Ethene'''
|----
|Structure
|[[Image:TS_with_labels.jpg|350px]]
|[[Image:Butadiene_with_labels.jpg|250px]]
|[[Image:Ethene_with_labels.jpg|250px]]
|----
|Bond length C6-C4/Å
|1.4
|1.45
|NA
|----
|Bond length C1-C4/Å
|1.38
|1.34
|NA
|----
|Bond length C6-C8/Å
|1.38
|1.34
|NA
|----
|Bond length C11-C12/Å
|1.38
|NA
|1.33
|----
|Bond angle (C4-C1-C12 & C6-C8-C11)/
|99.3
|
|NA
|----
|Bond forming (C6-C11)/Å
|2.12
|NA
|NA
|----
|Bond breaking (C1-C12)/Å
|2.12
|NA
|NA
|----
|}

From literature <ref name="0031-8949">F.H. Allen, Olga Kennard, D.G. Watson, ''J. Chem. Soc. Perkin Trans. II'', 1987, S1-S18 </ref><ref name="dewarcc">M. J. S. Dewar and H. N. Schmeising, ''Tetrahedron'', 1960, '''11''', {{DOI|10.1016/0040-4020(60)89012-6}} </ref>, a typical sp<sup>3</sup> C-C bond is 1.544Å long, while sp<sup>2</sup> C-C bond is 1.338Å long, and that the van der walls radius of carbon is 1.7Å. Judging from the data obtained, the partially formed bonds (2.12Å) are all longer than typical sp<sup>3</sup> C-C bond but shorter than twice the van der waals radius of carbon. It thus suggests that there is indeed bonding/overlap, and since a bond is still in the process of forming, it is an incomplete C-C bond, and hence, longer than usual.

===Molecular Orbitals===
The HOMO and LUMO orbitals of the transition states are shown in table 2.4:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.4: HOMO and LUMO of transition state''
|----
|
|'''HOMO'''
|'''LUMO'''
|----
|MO
|[[Image:YL_TS_HOMO.jpg|150px]]
|[[Image:YL_TS_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.32392
|0.02315
|----
|Symmetry (wrt plane)
|antisymmetric
|symmetric
|----
|}

Comparing to the HOMO and LUMO orbitals of the reactants shown earlier, it can be seen that the HOMO of transition state consists of the antisymmetric HOMO of cis-butadiene and the antisymmetric LUMO of ethane, forming a π(butadiene, diene)-π*(ethylene, dienophile) interaction, typical of cycloadditions. Since 6π electrons are involved (i.e. 4n+2 where n=1), under thermal conditions it would proceed via the Huckel route with all suprafacial components, and hence the bond forms on the same face of both the butadiene and ethylene π systems.

The HOMO(butadiene)-LUMO(ethylene) interaction is more favourable than the other way round due to the smaller energy gap:
*HOMO(butadiene)-LUMO(ethene) = 0.39665 Hartree (smaller, more favourable)
*HOMO(ethene)-LUMO(butadiene)= 0.40485 Hartree (larger, not favoured)

===Vibrational Frequency===
{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.5: Vibrations of Transition State''
|----
|'''Frequency/cm<sup>-1</sup>'''
|'''Description'''
|'''Mode of Vibration'''
|----
| -956
|Synchronous and concerted bond formation
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|147
|Rocking: ethene unit rocks about butadiene unit.
|[[Image:TS_vib_freq-147.jpg|150px]]
|----
|}

From the imaginary frequency of vibration, it can be seen that both bonds form in a concerted synchronous manner, where the ethene carbons move towards or away from butadiene unit together. This shows the concerted formation of bonds, as expected of a Diels Alder cycloaddition, as defined in the introduction of this section.

===Activation Energy===
The activation energy for the transition state is calculated at 298.15K at AM1 and at a higher level of B3LYP/6-31G* levels respectively for comparison. See table 2.6:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.6: Summary of energies''
|----
|
| colspan="2" | '''AM1'''
| colspan="2" | '''B3LYP/6-31G*'''
|-
|
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies '''
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies'''
|----
|'''Reactants (Cis-Butadiene+Ethene)''' /Hartree
|0.0750
|0.2594
| -234.5991
| -234.4299
|----
|'''Transition State''' /Hartree
|0.1116
|0.2594
| -234.5734
| -234.4876
|----
|'''Activation Energy E<sub>A</sub> /kcal mol<sup>-1</sup>
| --
|22.95
| --
|16.11
|----
|}

The energy differences between the transition state and ground state reactant are the activation energy, and only these differences can be compared (but not the absolute energies). It is found that the E<sub>A</sub> found at higher level of theory is smaller. However, both are of relatively similar orders of magnitude.

==Ex 3: Regioselectivity==

In this small exercise, the Diels-Alder cycloaddition between 1,3-Cyclohexadiene and Maleic Anhydride are investigated -- pertaining to its regioselectivity of exo or endo product. From literature, both theoretical and experimental, it is well known that endo-products are preferentially formed as the fast, kinetic product due to transition state stabilisation and thus lowered activation energy.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref><ref name="woodward hoffman">R. Hoffman, R. Woodward, ''J. Chem. Soc.'', 1965, '''87''', 4388-4389: {{DOI|10.1021/ja00947a033}}</ref> This will be verified here via computational predictions.

===Optimisation===
Both the endo and exo transition states are optimised using AM1 and B3LYP/6-31G* repectively, by optimising to TS(Berny). Summary to the calculations are in table 2.7.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.7: Comparison of optimisation of transition states''
|----
|
| colspan="2" | '''Exo-TS'''
| colspan="2" | '''Endo-TS'''
|----
|
|'''AM1'''
|'''B3LYP/6-31G*'''
|'''AM1'''
|'''B3LYP/6-31G*'''
|----
| Jmol
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Exo_TS.mol</uploadedFileContents></jmolApplet></jmol>
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Endo_TS.mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
| -0.0505
| -612.6793
| -0.0516
| -612.6833
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -811
| -448
| -806
| -446
|----
|Vibrations
| colspan="2" | [[Image:YL_Exo_vib.jpg|150px]]
| colspan="2" | [[Image:YL_Endo_vib.jpg|170px]]
|----
|}

As can be seen from above, endo transition state is more stable (i.e. lower in energy) than exo transition state, indicating that endo transition state is favoured – a lower transition state energy leads to reduced activation energy, and a preferred kinetic product.

Imaginary frequencies of exo and endo states show that the bond formation occurs in a concerted, synchronous manner, agreeing with the definition of Diels Alder cycloaddition definition.

===Molecular Orbitals===
HOMO and LUMO orbitals of both exo and endo TS are summaried in table 2.8 below:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.8: HOMO and LUMO of Exo- & Endo- TS''
|----
|
| colspan="2" | '''Exo TS'''
| colspan="2" | '''Endo TS'''
|----
|
|'''HOMO'''
|'''LUMO'''
|'''HOMO'''
|'''LUMO'''
|----
|Structure
|[[Image:YL_Exo_HOMO.jpg|150px]]
|[[Image:YL_Exo_LUMO.jpg|150px]]
|[[Image:YL_Endo_HOMO.jpg|150px]]
|[[Image:YL_Endo_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.3427
| -0.0404
| -0.3451
| -0.0357
|----
|Symmetry wrt Plane
|Antisymmetric
|Antisymmetric
|Antisymmetric
|Antisymmetric
|----
|}

All HOMO and LUMO orbitals of exo and endo TS are antisymmetric with respect to the mirror plane. Comparing with the HOMO and LUMOs of starting materials, it is observed that the transition state HOMOs consist of the interaction between HOMO of 1,3-cyclohexadiene and LUMO of maleic anhydride. Hence, this Diels Alder cycloaddition is between the HOMO of the diene and the LUMO of the dienophile.

===Geometries===

[[Image:Exo_geometry_labelled.jpg|380px]] [[Image:Endo_geometry_labelled.jpg|360px]]

Figure on the left: exo-TS. Figure on the right: endo-TS.

Note: All lengths are labelled in Å, all angles in degrees /°. All H atoms are not shown.

Since this is a concerted reaction, the breaking of maleic anhydride/cyclohexadiene bonds and forming of new σ-bonds are expected to be about the same length – which is indeed observed in both exo and endo isomers, where the forming/breaking length occurs at about 1.40Å (1.39 to 1.41Å).

It is noted that the distance between the maleic anhydride and the cyclohexadiene component above it is 2.95Å in exo isomer but 2.89Å in the endo isomer, suggesting that the steric crowding would cause endo-isomer to be less stable. However in contrary endo-TS is in fact more stable, and this phenomenon is explained by the stabilising overlap between π* (C=O) of maleic anhydride and π(C-C) of 1,3-cyclohexadiene, otherwise known as secondary orbital overlap, as demonstrated in the past by e.g. Fox et al <ref>M.A. Fox, R.Cardona, N.J. KIwiet, ''J. Org. Chem.'' 1987, '''52''', 1469-1474.{{DOI|10.1021/jo00384a016}}</ref>. This stabilises the whole system, and hence endo isomer is of a lower energy than its exo counterpart. It is to be noted that secondary orbital overlap occurs on top of the HOMO-LUMO primary orbital interaction of the frontier orbitals. A simplified analogous diagram below, by Fox et al, shows the stabilising interactions with secondary orbital contribution in endo isomer as compared to the exo-isomer.

[[Image:Secondary_orb_exo.jpg|300px]] [[Image:Secondary_orb_endo.jpg|300px]]

Other than these, most of the bond lengths are similar in both structures. More significant differences appear in the newly-formed bond angle between the two components -- 92.7° in exo-TS and 96.7° in endo-TS.

In conclusion, this exercise proves that, in agreement with literature, endo product is favoured in a Diels Alder reaction, due to its transition state being more stabilised (i.e. lower in energy) than its exo isomer, hence endo product is the kinetic product while exo is thermodynamic product.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref>

=Conclusions=

In conclusions, computational methods using Gaussian calculations (with various levels of theory and different approaches) can be used to predict the viability, stability and feasibility of transition states for a pericyclic reaction, eg cycloaddition (Diels-Alder) or sigmatropic Cope rearrangement. Both steric and electronic aspects can be studied by investigating energies, geometries, structures, frontier orbitals, and reaction coordinate/reaction pathways, providing a more holistic understanding of a reaction, especially if it is kinetically controlled by studying its possible transition states.

=References=
<references />

Rep:Mod:mariuspontmercy

2010-12-16T01:03:38Z

Ly308: /* Ex 1: HOMO & LUMO of Reactants */

'''''Note to Marker''': As most of the calculations are rather fast, they are mostly run on the laptop instead of on the college SCAN platform. As a result, most of these do not have a D-space link to the public repository. The major output files (eg .log or .chk) are hence uploaded as an "image" link and you can access it by clicking it/downloading it.''

=Cope Rearrangement=

Cope rearrangement is a percicyclic reaction of [3,3] sigmatropic nature, defined by its characteristic concerted migration of a group from one point of attachment to a conjugated system to another point of attachment. It involves the breaking and formation of one σ-bond each during the process. Here, the reactant, 1,5-hexadiene, is optimised to its minimum-energy (or most stable) conformation before a transition state is calculated. Intrinsic reaction coordinate (IRC) is finally used to relate the starting material/final product comformation to the transitions tate.

==Optimisation of Reactant==
===Conformers of 1,5-hexadiene: HF/3-21G===

1,5-Hexadiene was optimised with an “anti” linkage (i.e. antiperiplanar dihedral angle at its central four carbons), by using Hartree Fork (HF) and a basis set of 3-21G. The ''Anti-1'' conformer was obtained and symmetrised in Gaussview. Next, a gauche conformer (''Gauche3'') was obtained in a similar manner by altering its central dihedral angle. The results are summarized in table 1.1.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.1: Optimisation of 1,5-Hexadiene Conformers''
|----
|
|'''''Anti-1'''''
|'''''Anti-2'''''
|'''''Gauche-3'''''
|----
|Optimised Energy /Hartree
| -231.6926
| -231.6925
| -231.6927
|----
|Optimised Energy /kJ mol<sup>-1</sup>
| -614541
| -614541
| -614542
|----
|Point Group Symmetry
|C<sub>2</sub>
|C<sub>i</sub>
|C<sub>1</sub>
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Hexadiene_anti1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_anti2.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|----
|}

Counter intuitively, the ''gauche3'' conformer is more stable than the ''anti1'', even though the steric crowding in ''gauche3'' could lead to repulsions and destabilisation. Literature results, eg reported by Rocque et al <ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> have also famously proven that gauche conformers are more stable and preferred than anti ones. This is explained by the stereoelectronic effect, which involves stabilisating interaction between the π-electrons of the C=C and vinyl protons, otherwise known as CH-π interaction, thus stabilising ''gauche3'' instead.<ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> This is confirmed by performing another optimisation on an anti-conformer – ''anti2''. And indeed, it is observed that ''anti2'' and ''anti1'' are comparable in their energies, and ''gauche3'' is lower in energy.

To view optimisation files for anti1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI1.LOG

To view optimisation files for anti2: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI.2LOG

To view optimisation files for gauche3: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_GAUCHE3_OPT.LOG

===Optimisation of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

''Anti2'' conformer is optimised by B3LYP method, with basis set of 6-31G*. A summary and comparison of both methods (HF/3-21G and DFT/B3LYP/6-31G*) are shown in table 1.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.2: Comparison of optimisation properties of anti2 conformer, by different methods''
|----
|'''Method'''
|'''HF/3-21G'''
|'''DFT/B3LYP/6-31G*'''
|----
|Optimised Energy/Hartree
| -231.6925
| -234.6117
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614541
| -622284
|----
|'''Geometry'''
|[[Image:Hexadiene_anti2.jpg|400px]]
|[[Image:Hexadiene_anti2.jpg|400px]]
|----
|Bond lengths C1-C2 & C12-C13/Å
|1.33
|1.32
|----
|Bond lengths C2-C6 & C7-C12/Å
|1.50
|1.51
|----
|Bond length C6-C7/Å
|1.55
|1.55
|----
|Bond angles C1-C2-C6 & C7-C12-C13/°
|125.3
|124.8
|----
|Bond angles C2-C6-C7 & C6-C7-C12/°
|112.7
|111.4
|----
|Dihedral C1-C2-C6-C7 & C6-C7-C12-C13/°
|118.5
|114.7
|----
|Dihedral C2-C6-C7-C12/°
|180.0
|180.0
|----
|}

To view DFT optimisation file: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI2_DFT_opt.LOG
To view DFT freq file: {{DOI|10042/to-6215}}

As can be seen, the final energy calculated by DFT is lower than that by HF, and their geometry, though very similar, differs slightly e.g. dihedral angles at the sides, vinyl bond angles, and alkene bond angles.

===Thermochemistry of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

Thermochemistry data is obtained by doing a frequency calculation on the optimised structure, at the same level of theory DFT/B3LYP/6-31G*, at temperatures 0K and 298.15K respectively. Data is summarised in table 1.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.3: Comparison of Thermochemistry Data of Anti2 At 0K & 298.15K''
|----
|
|'''T=0K'''
|'''T=298.15K'''
|----
|Sum of electronic and zero point energies/Hartree
| -234.4663
| -234.4692
|----
|Sum of electronic and thermal energies/Hartree
| -234.4607
| -234.4619
|----
|Sum of electronic and thermal enthalpies/Hartree
| -234.4597
| -234.4609
|----
|Sum of electronic and thermal free energies/Hartree
| -234.4955
| -234.5008
|----
|}

To view opt+freq file at 0K: {{DOI|10042/to-6216}}

It is observed that the energies are higher for the system at 0K than at 298.15K, indicating that the molecule absorbs more heat at higher temperatures and thus thermal contribution to the energies increases.

==''Chair'' Product==

===Optimisation of Chair Product===

Two approaches are used to optimise the chair transition state structure, both at the level of theory of HF/3-21G, at default temperature of 198.15K and 1.0atm pressure. One was done using optimisation to TS(Berny) directly, calculating the force constant only once. The other was done using the redundant coordinate, first to freeze the breaking/forming bond at 2.2Å and optimise to minimum, then finally removing the freeze and optimise the whole structure to a minimum transition state TS(Berny). Results are summaried in table 1.4:

{| class="wikitable" border="1"
|+ ''Table 1.4: Comparison of Optimisation of Chair- Transition State with Different Conditions''
|----
|
|'''TS (Berny)'''<br />{{DOI|10042/to--6217}}
|'''TS (Berny) with Redundant<br />Coordinate''' <br />{{DOI|10042/to--6218}}<br />{{DOI|10042/to--6219}}
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_ts_TSBerny.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_TSBerny_(mod_red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Optimised Energy/Hartree
| -231.6193
| -231.6193
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614347.0313
| -614347.0313
|----
|Bond forming length/Å
|2.02043
|2.02069
|----
|Bond breaking length/Å
|2.02051
|2.02074
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -818
| -818
|----
|Vibration Figure
|[[Image:Chair_ts_HF_freq818.jpg|150px]]
|[[Image:Chair_ts_HF_freq818(mod-red).jpg|150px]]
|----
|}

Both sets of results yielded very similar outcomes, suggesting that either method is reliable and sufficient. Looking at the imaginary frequency at -818cm<sup>-1</sup>, it can be seen that the bond breaking and forming motions occur in a concerted manner simultaneously and asynchronously, corresponding to the definition of pericyclic sigmatropic reaction.

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

Intrinsic reaction coordinate (IRC) calculation was run to track the energy of the transition state along its reaction pathway, reaching a minimum value. The number of points along IRC was set at 100, and a minimum was reached successfully. This calculation is repeated for both force constant calculation for once and for always. See table 1.5

{| class="wikitable" border="1"
|+''Table 1.5: Comparison of IRC results for Chair-TS ''
|----
|'''Force Constant Computation'''
|'''Energy/Hartree'''
|'''Energy along IRC Path'''
|'''Gradient'''
|'''Minimum Structure'''
|----
|'''Once''' <br /> {{DOI|10042/to-6220}}
| -231.6851
|[[Image:Chair_IRC_(ONCE)_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_min_str.jpg|200px]]
|----
|'''Always'''<br /> {{DOI|10042/to-6221}}
| -231.6917
|[[Image:Chair_IRC_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_minimum_structure.jpg|200px]]
|----
|}

For the “force constant=once” calculation, even though IRC points set to 100, only 21 steps were performed by the programme and the energy graph did not plateau to a minimum, and its corresponding RMS gradient definitely did not reach zero. Hence, it did not successfully reach a minimum geometry. However, for the “force constant=always” calculation, the gradient has fallen to zero, it shows that the total energy of the structure has indeed fallen to a plateau minimum successfully, due to force constants being calculated at each and every step. This successful calculation can then be related to the gauche3 conformer obtained above, which has a comparable energy of -231.6927.

==Optimisation of ''Boat'' Product==

The boat transition state is optimised using yet another approach – with QST2 at the same level of theory. Here, it was ensured that the numbering of the atoms in reactant and that in product match exactly; otherwise, the calculation fails. Summary of optimisation is shown in table 1.6:

{| class="wikitable" border="1"
|+''Table 1.6: Optimisation results of Boat-Transition State by QST2''
|----
|Energy /Hartree
| -231.6028
|----
|Energy/kJ mol<sup>-1</sup>
| -614303.2667
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -840
|----
|Bond forming/breaking length/Å
|2.14
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Boat.mol</uploadedFileContents></jmolApplet></jmol>
|----
|D-space link
|{{DOI|10042/to-6223}}
|----
|}

Looking at the imaginary frequency at -840cm<sup>-1</sup>, the vibration motion stretches horizontally in plane in concerted and asynchronous manner, indicating again that the breaking and formation of bond is concerted and simultaneous, confirming the definition of Cope rearrangement.

==Activation Energy==

Thermochemical energies of chair/boat transition states are compared with reactant, and each corresponding activation energies for chair and boat pathways are calculated and compared, across T=0K and T=298.15K, all done at the same level of theory at DFT/B3LYP/6-31G*. The summary of energies are shown in table 1.7:

{| class="wikitable" border="1"
|+''Table 7: Comparison of Activation Energies of Chair & Boat TS at 0K & 298.15K''
|----
|
|
| colspan="4" | '''T=0K'''
| colspan="3" | '''T=298.15K'''
|----
|
| '''Electronic Energy'''
|'''Sum of electronic & zero-point energies/Hartree'''
|'''Activation Energy E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|'''E<sub>A</sub>(expt, 0K)/kcal mol<sup>-1</sup>'''
|'''Sum of electronic & thermal free energies/Hartree'''
|'''E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|----
|'''Chair TS'''
| -231.6193
| -234.4119
|0.0544
|34.11
|33.5+-0.5
| -234.4082
|0.0537
|33.67
|----
|'''Boat TS'''
| -231.6028
| -234.3990
|0.0673
|42.23
|44.7 +- 2.0
| -234.3960
|0.0659
|41.32
|----
|'''Reactant'''
| -231.6925
| -234.4663
|NA
|NA
|NA
| -234.4619
|NA
|NA
|----
|}

E<sub>A</sub> for T=0K is done by taking the difference between the sum of electronic & zero-point energies of reactant anti2 1,5-hexadiene and the chair/boat transition state, followed by converting the units from Hartree to kcal/mol (multiply by 627.509). E(A) for T=298.15K is done by the difference between their sum of electronic & thermal energies instead. The activation energy is found to be always higher for the boat than chair conformation, which agrees with expectation that chair-cyclohexane is usually the more stable form, with minimal steric clash due to the staggered H’s. In boat TS, the eclipsed hydrogens will tend to result in steric or torsional strain, causing destabilisation of the structure.

At T=0K, computational values found here are in good agreement with literature experimental values, with the values being just very slightly out of range from experimental data. This could be due to many trivial causes, especially since computational methods generate slightly different results each time it is run, or it could be due to the degree of accuracy imposed by the level of theory used. However, generally, the trend that chair-TS is more stable than its boat counterpart is well followed by computational methods here.

To view Chair-TS opt+freq by DFT:{{DOI|10042/to-6224}}{{DOI|10042/to-6225}} <br />
To view Boat-TS opt+freq by DFT: {{DOI|10042/to-6222}}{{DOI|10042/to-6226}}

=Diels Alder Reaction=

The Diels-Alder reaction is a [4+2] cycloaddition, between a conjugated diene and an dienophile, involving the concerted formation of two (or more) σ-bonds between the termini of two or more conjugated π-systems. The reaction investigated here is between cis-butadiene and ethylene; and the reaction between 1,3-cyclohexadiene and maleic anhydride is studied for its regioselectivity property.

==Ex 1: HOMO & LUMO of Reactants==

The HOMO and LUMO for reactants (cis-butadiene and ethylene) are obtained upon their optimised structure, done at the level of theory of semi-empirical/AM1. Their properties are summarised in table 2.1:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.1: HOMO & LUMO of cis-Butadiene & Ethylene (Reactants)''
|----
|
|'''Cis-Butadiene'''
|'''Ethylene'''
|----
|'''HOMO'''
|[[Image:YL_Butadiene_HOMO.jpg|150px]]
|[[Image:YL_Ethene_HOMO.jpg|150px]]
|----
|Energy /Hartree
| -0.34381
| -0.38777
|----
|Symmetry
|anti-symmetric
|symmetric
|----
|'''LUMO'''
|[[Image:YL_Butadiene_LUMO.jpg|150px]]
|[[Image:YL_Ehene_LUMO.jpg|150px]]
|----
|Energy /Hartree
|0.01708
|0.05284
|----
|Symmetry
|symmetric
|anti-symmetric
|----
|}

To view cis-butadiene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_MO.LOG <br />
To view ethylene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_MO.LOG

==Ex 2: Transition State Geometries==

===Optimisation/Geometry===

The transition state of the reaction is optimised with TS(Berny) with specified redundant coordinate method – the breaking/forming bonds are frozen to 2.1Å first and optimised to a minimum, before optimising with bond derivative to TS(Berny). The level of theory used is semiempirical/AM1. The optimisation summary shown in table 2.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.2: Optimisation of transition state''
|----
|Structure
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>TS_(mod-red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
|0.11163
|----
|Energy/kJ mol<sup>-1</sup>
|296
|----
|Bond forming length/Å
|2.1189
|----
|Bond breaking length/Å
|2.1196
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -956
|----
|Vibration
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|}

The optimised geometry is further investigated in table 2.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.3: Geometry of optimised transition state''
|----
|
|'''Transition State'''
|'''Butadiene'''
|'''Ethene'''
|----
|Structure
|[[Image:TS_with_labels.jpg|350px]]
|[[Image:Butadiene_with_labels.jpg|250px]]
|[[Image:Ethene_with_labels.jpg|250px]]
|----
|Bond length C6-C4/Å
|1.4
|1.45
|NA
|----
|Bond length C1-C4/Å
|1.38
|1.34
|NA
|----
|Bond length C6-C8/Å
|1.38
|1.34
|NA
|----
|Bond length C11-C12/Å
|1.38
|NA
|1.33
|----
|Bond angle (C4-C1-C12 & C6-C8-C11)/
|99.3
|
|NA
|----
|Bond forming (C6-C11)/Å
|2.12
|NA
|NA
|----
|Bond breaking (C1-C12)/Å
|2.12
|NA
|NA
|----
|}

From literature <ref name="0031-8949">F.H. Allen, Olga Kennard, D.G. Watson, ''J. Chem. Soc. Perkin Trans. II'', 1987, S1-S18 </ref><ref name="dewarcc">M. J. S. Dewar and H. N. Schmeising, ''Tetrahedron'', 1960, '''11''', {{DOI|10.1016/0040-4020(60)89012-6}} </ref>, a typical sp<sup>3</sup> C-C bond is 1.544Å long, while sp<sup>2</sup> C-C bond is 1.338Å long, and that the van der walls radius of carbon is 1.7Å. Judging from the data obtained, the partially formed bonds (2.12Å) are all longer than typical sp<sup>3</sup> C-C bond but shorter than twice the van der waals radius of carbon. It thus suggests that there is indeed bonding/overlap, and since a bond is still in the process of forming, it is an incomplete C-C bond, and hence, longer than usual.

===Molecular Orbitals===
The HOMO and LUMO orbitals of the transition states are shown in table 2.4:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.4: HOMO and LUMO of transition state''
|----
|
|'''HOMO'''
|'''LUMO'''
|----
|MO
|[[Image:YL_TS_HOMO.jpg|150px]]
|[[Image:YL_TS_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.32392
|0.02315
|----
|Symmetry (wrt plane)
|antisymmetric
|symmetric
|----
|}

Comparing to the HOMO and LUMO orbitals of the reactants shown earlier, it can be seen that the HOMO of transition state consists of the antisymmetric HOMO of cis-butadiene and the antisymmetric LUMO of ethane, forming a π(butadiene, diene)-π*(ethylene, dienophile) interaction, typical of cycloadditions. Since 6π electrons are involved (i.e. 4n+2 where n=1), under thermal conditions it would proceed via the Huckel route with all suprafacial components, and hence the bond forms on the same face of both the butadiene and ethylene π systems.

The HOMO(butadiene)-LUMO(ethylene) interaction is more favourable than the other way round due to the smaller energy gap:
*HOMO(butadiene)-LUMO(ethene) = 0.39665 Hartree (smaller, more favourable)
*HOMO(ethene)-LUMO(butadiene)= 0.40485 Hartree (larger, not favoured)

===Vibrational Frequency===
{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.5: Vibrations of Transition State''
|----
|'''Frequency/cm<sup>-1</sup>'''
|'''Description'''
|'''Mode of Vibration'''
|----
| -956
|Synchronous and concerted bond formation
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|147
|Rocking: ethene unit rocks about butadiene unit.
|[[Image:TS_vib_freq-147.jpg|150px]]
|----
|}

From the imaginary frequency of vibration, it can be seen that both bonds form in a concerted synchronous manner, where the ethene carbons move towards or away from butadiene unit together. This shows the concerted formation of bonds, as expected of a Diels Alder cycloaddition, as defined in the introduction of this section.

===Activation Energy===
The activation energy for the transition state is calculated at 298.15K at AM1 and at a higher level of B3LYP/6-31G* levels respectively for comparison. See table 2.6:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.6: Summary of energies''
|----
|
| colspan="2" | '''AM1'''
| colspan="2" | '''B3LYP/6-31G*'''
|-
|
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies '''
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies'''
|----
|'''Reactants (Cis-Butadiene+Ethene)''' /Hartree
|0.0750
|0.2594
| -234.5991
| -234.4299
|----
|'''Transition State''' /Hartree
|0.1116
|0.2594
| -234.5734
| -234.4876
|----
|'''Activation Energy E<sub>A</sub> /kcal mol<sup>-1</sup>
| --
|22.95
| --
|16.11
|----
|}

The energy differences between the transition state and ground state reactant are the activation energy, and only these differences can be compared (but not the absolute energies). It is found that the E<sub>A</sub> found at higher level of theory is smaller. However, both are of relatively similar orders of magnitude.

==Ex 3: Regioselectivity==

In this small exercise, the Diels-Alder cycloaddition between 1,3-Cyclohexadiene and Maleic Anhydride are investigated -- pertaining to its regioselectivity of exo or endo product. From literature, both theoretical and experimental, it is well known that endo-products are preferentially formed as the fast, kinetic product due to transition state stabilisation and thus lowered activation energy.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref><ref name="woodward hoffman">R. Hoffman, R. Woodward, ''J. Chem. Soc.'', 1965, '''87''', 4388-4389: {{DOI|10.1021/ja00947a033}}</ref> This will be verified here via computational predictions.

===Optimisation===
Both the endo and exo transition states are optimised using AM1 and B3LYP/6-31G* repectively, by optimising to TS(Berny). Summary to the calculations are in table 2.7.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.7: Comparison of optimisation of transition states''
|----
|
| colspan="2" | '''Exo-TS'''
| colspan="2" | '''Endo-TS'''
|----
|
|'''AM1'''
|'''B3LYP/6-31G*'''
|'''AM1'''
|'''B3LYP/6-31G*'''
|----
| Jmol
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Exo_TS.mol</uploadedFileContents></jmolApplet></jmol>
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Endo_TS.mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
| -0.0505
| -612.6793
| -0.0516
| -612.6833
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -811
| -448
| -806
| -446
|----
|Vibrations
| colspan="2" | [[Image:YL_Exo_vib.jpg|150px]]
| colspan="2" | [[Image:YL_Endo_vib.jpg|170px]]
|----
|}

As can be seen from above, endo transition state is more stable (i.e. lower in energy) than exo transition state, indicating that endo transition state is favoured – a lower transition state energy leads to reduced activation energy, and a preferred kinetic product.

Imaginary frequencies of exo and endo states show that the bond formation occurs in a concerted, synchronous manner, agreeing with the definition of Diels Alder cycloaddition definition.

===Molecular Orbitals===
HOMO and LUMO orbitals of both exo and endo TS are summaried in table 2.8 below:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.8: HOMO and LUMO of Exo- & Endo- TS''
|----
|
| colspan="2" | '''Exo TS'''
| colspan="2" | '''Endo TS'''
|----
|
|'''HOMO'''
|'''LUMO'''
|'''HOMO'''
|'''LUMO'''
|----
|Structure
|[[Image:YL_Exo_HOMO.jpg|150px]]
|[[Image:YL_Exo_LUMO.jpg|150px]]
|[[Image:YL_Endo_HOMO.jpg|150px]]
|[[Image:YL_Endo_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.3427
| -0.0404
| -0.3451
| -0.0357
|----
|Symmetry wrt Plane
|Antisymmetric
|Antisymmetric
|Antisymmetric
|Antisymmetric
|----
|}

All HOMO and LUMO orbitals of exo and endo TS are antisymmetric with respect to the mirror plane. Comparing with the HOMO and LUMOs of starting materials, it is observed that the transition state HOMOs consist of the interaction between HOMO of 1,3-cyclohexadiene and LUMO of maleic anhydride. Hence, this Diels Alder cycloaddition is between the HOMO of the diene and the LUMO of the dienophile.

===Geometries===

[[Image:Exo_geometry_labelled.jpg|380px]] [[Image:Endo_geometry_labelled.jpg|360px]]

Figure on the left: exo-TS. Figure on the right: endo-TS.

Note: All lengths are labelled in Å, all angles in degrees /°. All H atoms are not shown.

Since this is a concerted reaction, the breaking of maleic anhydride/cyclohexadiene bonds and forming of new σ-bonds are expected to be about the same length – which is indeed observed in both exo and endo isomers, where the forming/breaking length occurs at about 1.40Å (1.39 to 1.41Å).

It is noted that the distance between the maleic anhydride and the cyclohexadiene component above it is 2.95Å in exo isomer but 2.89Å in the endo isomer, suggesting that the steric crowding would cause endo-isomer to be less stable. However in contrary endo-TS is in fact more stable, and this phenomenon is explained by the stabilising overlap between π* (C=O) of maleic anhydride and π(C-C) of 1,3-cyclohexadiene, otherwise known as secondary orbital overlap, as demonstrated in the past by e.g. Fox et al <ref>M.A. Fox, R.Cardona, N.J. KIwiet, ''J. Org. Chem.'' 1987, '''52''', 1469-1474.{{DOI|10.1021/jo00384a016}}</ref>. This stabilises the whole system, and hence endo isomer is of a lower energy than its exo counterpart. It is to be noted that secondary orbital overlap occurs on top of the HOMO-LUMO primary orbital interaction of the frontier orbitals. A simplified analogous diagram below, by Fox et al, shows the stabilising interactions with secondary orbital contribution in endo isomer as compared to the exo-isomer.

[[Image:Secondary_orb_exo.jpg|300px]] [[Image:Secondary_orb_endo.jpg|300px]]

Other than these, most of the bond lengths are similar in both structures. More significant differences appear in the newly-formed bond angle between the two components -- 92.7° in exo-TS and 96.7° in endo-TS.

In conclusion, this exercise proves that, in agreement with literature, endo product is favoured in a Diels Alder reaction, due to its transition state being more stabilised (i.e. lower in energy) than its exo isomer, hence endo product is the kinetic product while exo is thermodynamic product.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref>

=Conclusions=

In conclusions, computational methods using Gaussian calculations (with various levels of theory and different approaches) can be used to predict the viability, stability and feasibility of transition states for a pericyclic reaction, eg cycloaddition (Diels-Alder) or sigmatropic Cope rearrangement. Both steric and electronic aspects can be studied by investigating energies, geometries, structures, frontier orbitals, and reaction coordinate/reaction pathways, providing a more holistic understanding of a reaction, especially if it is kinetically controlled by studying its possible transition states.

=References=
<references />

Rep:Mod:mariuspontmercy

2010-12-16T01:03:16Z

Ly308: /* Ex 1: HOMO & LUMO of Reactants */

'''''Note to Marker''': As most of the calculations are rather fast, they are mostly run on the laptop instead of on the college SCAN platform. As a result, most of these do not have a D-space link to the public repository. The major output files (eg .log or .chk) are hence uploaded as an "image" link and you can access it by clicking it/downloading it.''

=Cope Rearrangement=

Cope rearrangement is a percicyclic reaction of [3,3] sigmatropic nature, defined by its characteristic concerted migration of a group from one point of attachment to a conjugated system to another point of attachment. It involves the breaking and formation of one σ-bond each during the process. Here, the reactant, 1,5-hexadiene, is optimised to its minimum-energy (or most stable) conformation before a transition state is calculated. Intrinsic reaction coordinate (IRC) is finally used to relate the starting material/final product comformation to the transitions tate.

==Optimisation of Reactant==
===Conformers of 1,5-hexadiene: HF/3-21G===

1,5-Hexadiene was optimised with an “anti” linkage (i.e. antiperiplanar dihedral angle at its central four carbons), by using Hartree Fork (HF) and a basis set of 3-21G. The ''Anti-1'' conformer was obtained and symmetrised in Gaussview. Next, a gauche conformer (''Gauche3'') was obtained in a similar manner by altering its central dihedral angle. The results are summarized in table 1.1.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.1: Optimisation of 1,5-Hexadiene Conformers''
|----
|
|'''''Anti-1'''''
|'''''Anti-2'''''
|'''''Gauche-3'''''
|----
|Optimised Energy /Hartree
| -231.6926
| -231.6925
| -231.6927
|----
|Optimised Energy /kJ mol<sup>-1</sup>
| -614541
| -614541
| -614542
|----
|Point Group Symmetry
|C<sub>2</sub>
|C<sub>i</sub>
|C<sub>1</sub>
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Hexadiene_anti1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_anti2.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Hexadiene_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|----
|}

Counter intuitively, the ''gauche3'' conformer is more stable than the ''anti1'', even though the steric crowding in ''gauche3'' could lead to repulsions and destabilisation. Literature results, eg reported by Rocque et al <ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> have also famously proven that gauche conformers are more stable and preferred than anti ones. This is explained by the stereoelectronic effect, which involves stabilisating interaction between the π-electrons of the C=C and vinyl protons, otherwise known as CH-π interaction, thus stabilising ''gauche3'' instead.<ref name="Rocque">B. G. Rocque, J. M. Gonzales, H. F. Schaefer III, ''Molecular Phy.'', 2002, '''100''', 441 - 446 {{DOI|10.1080/00268970110081412}}</ref> This is confirmed by performing another optimisation on an anti-conformer – ''anti2''. And indeed, it is observed that ''anti2'' and ''anti1'' are comparable in their energies, and ''gauche3'' is lower in energy.

To view optimisation files for anti1: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI1.LOG

To view optimisation files for anti2: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI.2LOG

To view optimisation files for gauche3: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_GAUCHE3_OPT.LOG

===Optimisation of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

''Anti2'' conformer is optimised by B3LYP method, with basis set of 6-31G*. A summary and comparison of both methods (HF/3-21G and DFT/B3LYP/6-31G*) are shown in table 1.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.2: Comparison of optimisation properties of anti2 conformer, by different methods''
|----
|'''Method'''
|'''HF/3-21G'''
|'''DFT/B3LYP/6-31G*'''
|----
|Optimised Energy/Hartree
| -231.6925
| -234.6117
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614541
| -622284
|----
|'''Geometry'''
|[[Image:Hexadiene_anti2.jpg|400px]]
|[[Image:Hexadiene_anti2.jpg|400px]]
|----
|Bond lengths C1-C2 & C12-C13/Å
|1.33
|1.32
|----
|Bond lengths C2-C6 & C7-C12/Å
|1.50
|1.51
|----
|Bond length C6-C7/Å
|1.55
|1.55
|----
|Bond angles C1-C2-C6 & C7-C12-C13/°
|125.3
|124.8
|----
|Bond angles C2-C6-C7 & C6-C7-C12/°
|112.7
|111.4
|----
|Dihedral C1-C2-C6-C7 & C6-C7-C12-C13/°
|118.5
|114.7
|----
|Dihedral C2-C6-C7-C12/°
|180.0
|180.0
|----
|}

To view DFT optimisation file: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:HEXADIENE_ANTI2_DFT_opt.LOG
To view DFT freq file: {{DOI|10042/to-6215}}

As can be seen, the final energy calculated by DFT is lower than that by HF, and their geometry, though very similar, differs slightly e.g. dihedral angles at the sides, vinyl bond angles, and alkene bond angles.

===Thermochemistry of ''Anti-2'': HF/3-21G & DFT/B3LYP/6-31G(d)===

Thermochemistry data is obtained by doing a frequency calculation on the optimised structure, at the same level of theory DFT/B3LYP/6-31G*, at temperatures 0K and 298.15K respectively. Data is summarised in table 1.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 1.3: Comparison of Thermochemistry Data of Anti2 At 0K & 298.15K''
|----
|
|'''T=0K'''
|'''T=298.15K'''
|----
|Sum of electronic and zero point energies/Hartree
| -234.4663
| -234.4692
|----
|Sum of electronic and thermal energies/Hartree
| -234.4607
| -234.4619
|----
|Sum of electronic and thermal enthalpies/Hartree
| -234.4597
| -234.4609
|----
|Sum of electronic and thermal free energies/Hartree
| -234.4955
| -234.5008
|----
|}

To view opt+freq file at 0K: {{DOI|10042/to-6216}}

It is observed that the energies are higher for the system at 0K than at 298.15K, indicating that the molecule absorbs more heat at higher temperatures and thus thermal contribution to the energies increases.

==''Chair'' Product==

===Optimisation of Chair Product===

Two approaches are used to optimise the chair transition state structure, both at the level of theory of HF/3-21G, at default temperature of 198.15K and 1.0atm pressure. One was done using optimisation to TS(Berny) directly, calculating the force constant only once. The other was done using the redundant coordinate, first to freeze the breaking/forming bond at 2.2Å and optimise to minimum, then finally removing the freeze and optimise the whole structure to a minimum transition state TS(Berny). Results are summaried in table 1.4:

{| class="wikitable" border="1"
|+ ''Table 1.4: Comparison of Optimisation of Chair- Transition State with Different Conditions''
|----
|
|'''TS (Berny)'''<br />{{DOI|10042/to--6217}}
|'''TS (Berny) with Redundant<br />Coordinate''' <br />{{DOI|10042/to--6218}}<br />{{DOI|10042/to--6219}}
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_ts_TSBerny.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>Chair_TSBerny_(mod_red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Optimised Energy/Hartree
| -231.6193
| -231.6193
|----
|Optimised Energy/kJ mol<sup>-1</sup>
| -614347.0313
| -614347.0313
|----
|Bond forming length/Å
|2.02043
|2.02069
|----
|Bond breaking length/Å
|2.02051
|2.02074
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -818
| -818
|----
|Vibration Figure
|[[Image:Chair_ts_HF_freq818.jpg|150px]]
|[[Image:Chair_ts_HF_freq818(mod-red).jpg|150px]]
|----
|}

Both sets of results yielded very similar outcomes, suggesting that either method is reliable and sufficient. Looking at the imaginary frequency at -818cm<sup>-1</sup>, it can be seen that the bond breaking and forming motions occur in a concerted manner simultaneously and asynchronously, corresponding to the definition of pericyclic sigmatropic reaction.

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

Intrinsic reaction coordinate (IRC) calculation was run to track the energy of the transition state along its reaction pathway, reaching a minimum value. The number of points along IRC was set at 100, and a minimum was reached successfully. This calculation is repeated for both force constant calculation for once and for always. See table 1.5

{| class="wikitable" border="1"
|+''Table 1.5: Comparison of IRC results for Chair-TS ''
|----
|'''Force Constant Computation'''
|'''Energy/Hartree'''
|'''Energy along IRC Path'''
|'''Gradient'''
|'''Minimum Structure'''
|----
|'''Once''' <br /> {{DOI|10042/to-6220}}
| -231.6851
|[[Image:Chair_IRC_(ONCE)_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_(ONCE)_min_str.jpg|200px]]
|----
|'''Always'''<br /> {{DOI|10042/to-6221}}
| -231.6917
|[[Image:Chair_IRC_energy_graph.jpg|250px]]
|[[Image:Chair_IRC_grad_graph.jpg|250px]]
|[[Image:Chair_IRC_minimum_structure.jpg|200px]]
|----
|}

For the “force constant=once” calculation, even though IRC points set to 100, only 21 steps were performed by the programme and the energy graph did not plateau to a minimum, and its corresponding RMS gradient definitely did not reach zero. Hence, it did not successfully reach a minimum geometry. However, for the “force constant=always” calculation, the gradient has fallen to zero, it shows that the total energy of the structure has indeed fallen to a plateau minimum successfully, due to force constants being calculated at each and every step. This successful calculation can then be related to the gauche3 conformer obtained above, which has a comparable energy of -231.6927.

==Optimisation of ''Boat'' Product==

The boat transition state is optimised using yet another approach – with QST2 at the same level of theory. Here, it was ensured that the numbering of the atoms in reactant and that in product match exactly; otherwise, the calculation fails. Summary of optimisation is shown in table 1.6:

{| class="wikitable" border="1"
|+''Table 1.6: Optimisation results of Boat-Transition State by QST2''
|----
|Energy /Hartree
| -231.6028
|----
|Energy/kJ mol<sup>-1</sup>
| -614303.2667
|----
|Imaginary Vibrational Frequency/cm<sup>-1</sup>
| -840
|----
|Bond forming/breaking length/Å
|2.14
|----
|Jmol
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Boat.mol</uploadedFileContents></jmolApplet></jmol>
|----
|D-space link
|{{DOI|10042/to-6223}}
|----
|}

Looking at the imaginary frequency at -840cm<sup>-1</sup>, the vibration motion stretches horizontally in plane in concerted and asynchronous manner, indicating again that the breaking and formation of bond is concerted and simultaneous, confirming the definition of Cope rearrangement.

==Activation Energy==

Thermochemical energies of chair/boat transition states are compared with reactant, and each corresponding activation energies for chair and boat pathways are calculated and compared, across T=0K and T=298.15K, all done at the same level of theory at DFT/B3LYP/6-31G*. The summary of energies are shown in table 1.7:

{| class="wikitable" border="1"
|+''Table 7: Comparison of Activation Energies of Chair & Boat TS at 0K & 298.15K''
|----
|
|
| colspan="4" | '''T=0K'''
| colspan="3" | '''T=298.15K'''
|----
|
| '''Electronic Energy'''
|'''Sum of electronic & zero-point energies/Hartree'''
|'''Activation Energy E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|'''E<sub>A</sub>(expt, 0K)/kcal mol<sup>-1</sup>'''
|'''Sum of electronic & thermal free energies/Hartree'''
|'''E<sub>A</sub>/Hartree'''
|'''E<sub>A</sub>/kcal mol<sup>-1</sup>'''
|----
|'''Chair TS'''
| -231.6193
| -234.4119
|0.0544
|34.11
|33.5+-0.5
| -234.4082
|0.0537
|33.67
|----
|'''Boat TS'''
| -231.6028
| -234.3990
|0.0673
|42.23
|44.7 +- 2.0
| -234.3960
|0.0659
|41.32
|----
|'''Reactant'''
| -231.6925
| -234.4663
|NA
|NA
|NA
| -234.4619
|NA
|NA
|----
|}

E<sub>A</sub> for T=0K is done by taking the difference between the sum of electronic & zero-point energies of reactant anti2 1,5-hexadiene and the chair/boat transition state, followed by converting the units from Hartree to kcal/mol (multiply by 627.509). E(A) for T=298.15K is done by the difference between their sum of electronic & thermal energies instead. The activation energy is found to be always higher for the boat than chair conformation, which agrees with expectation that chair-cyclohexane is usually the more stable form, with minimal steric clash due to the staggered H’s. In boat TS, the eclipsed hydrogens will tend to result in steric or torsional strain, causing destabilisation of the structure.

At T=0K, computational values found here are in good agreement with literature experimental values, with the values being just very slightly out of range from experimental data. This could be due to many trivial causes, especially since computational methods generate slightly different results each time it is run, or it could be due to the degree of accuracy imposed by the level of theory used. However, generally, the trend that chair-TS is more stable than its boat counterpart is well followed by computational methods here.

To view Chair-TS opt+freq by DFT:{{DOI|10042/to-6224}}{{DOI|10042/to-6225}} <br />
To view Boat-TS opt+freq by DFT: {{DOI|10042/to-6222}}{{DOI|10042/to-6226}}

=Diels Alder Reaction=

The Diels-Alder reaction is a [4+2] cycloaddition, between a conjugated diene and an dienophile, involving the concerted formation of two (or more) σ-bonds between the termini of two or more conjugated π-systems. The reaction investigated here is between cis-butadiene and ethylene; and the reaction between 1,3-cyclohexadiene and maleic anhydride is studied for its regioselectivity property.

==Ex 1: HOMO & LUMO of Reactants==

The HOMO and LUMO for reactants (cis-butadiene and ethylene) are obtained upon their optimised structure, done at the level of theory of semi-empirical/AM1. Their properties are summarised in table 2.1:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.1: HOMO & LUMO of cis-Butadiene & Ethylene (Reactants)''
|----
|
|'''Cis-Butadiene'''
|'''Ethylene'''
|----
|'''HOMO'''
|[[Image:YL_Butadiene_HOMO.jpg|150px]]
|[[Image:YL_Ethene_HOMO.jpg|150px]]
|----
|Energy /Hartree
| -0.34381
| -0.38777
|----
|Symmetry
|anti-symmetric
|symmetric
|----
|'''LUMO'''
|[[Image:YL_Butadiene_LUMO.jpg|150px]]
|[[Image:YL_Ehene_LUMO.jpg|150px]]
|----
|Energy /Hartree
|0.01708
|0.05284
|----
|Symmetry
|symmetric
|anti-symmetric
|----
|}

To view cis-butadiene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:BUTADIENE_MO.LOG
To view ethylene optimisation/MO calculation: https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:ETHENE_MO.LOG

==Ex 2: Transition State Geometries==

===Optimisation/Geometry===

The transition state of the reaction is optimised with TS(Berny) with specified redundant coordinate method – the breaking/forming bonds are frozen to 2.1Å first and optimised to a minimum, before optimising with bond derivative to TS(Berny). The level of theory used is semiempirical/AM1. The optimisation summary shown in table 2.2:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.2: Optimisation of transition state''
|----
|Structure
|<jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>TS_(mod-red).mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
|0.11163
|----
|Energy/kJ mol<sup>-1</sup>
|296
|----
|Bond forming length/Å
|2.1189
|----
|Bond breaking length/Å
|2.1196
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -956
|----
|Vibration
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|}

The optimised geometry is further investigated in table 2.3:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.3: Geometry of optimised transition state''
|----
|
|'''Transition State'''
|'''Butadiene'''
|'''Ethene'''
|----
|Structure
|[[Image:TS_with_labels.jpg|350px]]
|[[Image:Butadiene_with_labels.jpg|250px]]
|[[Image:Ethene_with_labels.jpg|250px]]
|----
|Bond length C6-C4/Å
|1.4
|1.45
|NA
|----
|Bond length C1-C4/Å
|1.38
|1.34
|NA
|----
|Bond length C6-C8/Å
|1.38
|1.34
|NA
|----
|Bond length C11-C12/Å
|1.38
|NA
|1.33
|----
|Bond angle (C4-C1-C12 & C6-C8-C11)/
|99.3
|
|NA
|----
|Bond forming (C6-C11)/Å
|2.12
|NA
|NA
|----
|Bond breaking (C1-C12)/Å
|2.12
|NA
|NA
|----
|}

From literature <ref name="0031-8949">F.H. Allen, Olga Kennard, D.G. Watson, ''J. Chem. Soc. Perkin Trans. II'', 1987, S1-S18 </ref><ref name="dewarcc">M. J. S. Dewar and H. N. Schmeising, ''Tetrahedron'', 1960, '''11''', {{DOI|10.1016/0040-4020(60)89012-6}} </ref>, a typical sp<sup>3</sup> C-C bond is 1.544Å long, while sp<sup>2</sup> C-C bond is 1.338Å long, and that the van der walls radius of carbon is 1.7Å. Judging from the data obtained, the partially formed bonds (2.12Å) are all longer than typical sp<sup>3</sup> C-C bond but shorter than twice the van der waals radius of carbon. It thus suggests that there is indeed bonding/overlap, and since a bond is still in the process of forming, it is an incomplete C-C bond, and hence, longer than usual.

===Molecular Orbitals===
The HOMO and LUMO orbitals of the transition states are shown in table 2.4:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.4: HOMO and LUMO of transition state''
|----
|
|'''HOMO'''
|'''LUMO'''
|----
|MO
|[[Image:YL_TS_HOMO.jpg|150px]]
|[[Image:YL_TS_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.32392
|0.02315
|----
|Symmetry (wrt plane)
|antisymmetric
|symmetric
|----
|}

Comparing to the HOMO and LUMO orbitals of the reactants shown earlier, it can be seen that the HOMO of transition state consists of the antisymmetric HOMO of cis-butadiene and the antisymmetric LUMO of ethane, forming a π(butadiene, diene)-π*(ethylene, dienophile) interaction, typical of cycloadditions. Since 6π electrons are involved (i.e. 4n+2 where n=1), under thermal conditions it would proceed via the Huckel route with all suprafacial components, and hence the bond forms on the same face of both the butadiene and ethylene π systems.

The HOMO(butadiene)-LUMO(ethylene) interaction is more favourable than the other way round due to the smaller energy gap:
*HOMO(butadiene)-LUMO(ethene) = 0.39665 Hartree (smaller, more favourable)
*HOMO(ethene)-LUMO(butadiene)= 0.40485 Hartree (larger, not favoured)

===Vibrational Frequency===
{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.5: Vibrations of Transition State''
|----
|'''Frequency/cm<sup>-1</sup>'''
|'''Description'''
|'''Mode of Vibration'''
|----
| -956
|Synchronous and concerted bond formation
|[[Image:TS_vib_freq-956.jpg|150px]]
|----
|147
|Rocking: ethene unit rocks about butadiene unit.
|[[Image:TS_vib_freq-147.jpg|150px]]
|----
|}

From the imaginary frequency of vibration, it can be seen that both bonds form in a concerted synchronous manner, where the ethene carbons move towards or away from butadiene unit together. This shows the concerted formation of bonds, as expected of a Diels Alder cycloaddition, as defined in the introduction of this section.

===Activation Energy===
The activation energy for the transition state is calculated at 298.15K at AM1 and at a higher level of B3LYP/6-31G* levels respectively for comparison. See table 2.6:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.6: Summary of energies''
|----
|
| colspan="2" | '''AM1'''
| colspan="2" | '''B3LYP/6-31G*'''
|-
|
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies '''
|'''Electronic Energy '''
|'''Sum of electronic & thermal energies'''
|----
|'''Reactants (Cis-Butadiene+Ethene)''' /Hartree
|0.0750
|0.2594
| -234.5991
| -234.4299
|----
|'''Transition State''' /Hartree
|0.1116
|0.2594
| -234.5734
| -234.4876
|----
|'''Activation Energy E<sub>A</sub> /kcal mol<sup>-1</sup>
| --
|22.95
| --
|16.11
|----
|}

The energy differences between the transition state and ground state reactant are the activation energy, and only these differences can be compared (but not the absolute energies). It is found that the E<sub>A</sub> found at higher level of theory is smaller. However, both are of relatively similar orders of magnitude.

==Ex 3: Regioselectivity==

In this small exercise, the Diels-Alder cycloaddition between 1,3-Cyclohexadiene and Maleic Anhydride are investigated -- pertaining to its regioselectivity of exo or endo product. From literature, both theoretical and experimental, it is well known that endo-products are preferentially formed as the fast, kinetic product due to transition state stabilisation and thus lowered activation energy.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref><ref name="woodward hoffman">R. Hoffman, R. Woodward, ''J. Chem. Soc.'', 1965, '''87''', 4388-4389: {{DOI|10.1021/ja00947a033}}</ref> This will be verified here via computational predictions.

===Optimisation===
Both the endo and exo transition states are optimised using AM1 and B3LYP/6-31G* repectively, by optimising to TS(Berny). Summary to the calculations are in table 2.7.

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.7: Comparison of optimisation of transition states''
|----
|
| colspan="2" | '''Exo-TS'''
| colspan="2" | '''Endo-TS'''
|----
|
|'''AM1'''
|'''B3LYP/6-31G*'''
|'''AM1'''
|'''B3LYP/6-31G*'''
|----
| Jmol
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Exo_TS.mol</uploadedFileContents></jmolApplet></jmol>
| colspan="2" | <jmol><jmolApplet><size>150</size><script>cpk -20;</script><uploadedFileContents>YL_Endo_TS.mol</uploadedFileContents></jmolApplet></jmol>
|----
|Energy/Hartree
| -0.0505
| -612.6793
| -0.0516
| -612.6833
|----
|Imaginary Frequency/cm<sup>-1</sup>
| -811
| -448
| -806
| -446
|----
|Vibrations
| colspan="2" | [[Image:YL_Exo_vib.jpg|150px]]
| colspan="2" | [[Image:YL_Endo_vib.jpg|170px]]
|----
|}

As can be seen from above, endo transition state is more stable (i.e. lower in energy) than exo transition state, indicating that endo transition state is favoured – a lower transition state energy leads to reduced activation energy, and a preferred kinetic product.

Imaginary frequencies of exo and endo states show that the bond formation occurs in a concerted, synchronous manner, agreeing with the definition of Diels Alder cycloaddition definition.

===Molecular Orbitals===
HOMO and LUMO orbitals of both exo and endo TS are summaried in table 2.8 below:

{| class="wikitable" border="1"
|- align="center"
|+ ''Table 2.8: HOMO and LUMO of Exo- & Endo- TS''
|----
|
| colspan="2" | '''Exo TS'''
| colspan="2" | '''Endo TS'''
|----
|
|'''HOMO'''
|'''LUMO'''
|'''HOMO'''
|'''LUMO'''
|----
|Structure
|[[Image:YL_Exo_HOMO.jpg|150px]]
|[[Image:YL_Exo_LUMO.jpg|150px]]
|[[Image:YL_Endo_HOMO.jpg|150px]]
|[[Image:YL_Endo_LUMO.jpg|150px]]
|----
|Energy/Hartree
| -0.3427
| -0.0404
| -0.3451
| -0.0357
|----
|Symmetry wrt Plane
|Antisymmetric
|Antisymmetric
|Antisymmetric
|Antisymmetric
|----
|}

All HOMO and LUMO orbitals of exo and endo TS are antisymmetric with respect to the mirror plane. Comparing with the HOMO and LUMOs of starting materials, it is observed that the transition state HOMOs consist of the interaction between HOMO of 1,3-cyclohexadiene and LUMO of maleic anhydride. Hence, this Diels Alder cycloaddition is between the HOMO of the diene and the LUMO of the dienophile.

===Geometries===

[[Image:Exo_geometry_labelled.jpg|380px]] [[Image:Endo_geometry_labelled.jpg|360px]]

Figure on the left: exo-TS. Figure on the right: endo-TS.

Note: All lengths are labelled in Å, all angles in degrees /°. All H atoms are not shown.

Since this is a concerted reaction, the breaking of maleic anhydride/cyclohexadiene bonds and forming of new σ-bonds are expected to be about the same length – which is indeed observed in both exo and endo isomers, where the forming/breaking length occurs at about 1.40Å (1.39 to 1.41Å).

It is noted that the distance between the maleic anhydride and the cyclohexadiene component above it is 2.95Å in exo isomer but 2.89Å in the endo isomer, suggesting that the steric crowding would cause endo-isomer to be less stable. However in contrary endo-TS is in fact more stable, and this phenomenon is explained by the stabilising overlap between π* (C=O) of maleic anhydride and π(C-C) of 1,3-cyclohexadiene, otherwise known as secondary orbital overlap, as demonstrated in the past by e.g. Fox et al <ref>M.A. Fox, R.Cardona, N.J. KIwiet, ''J. Org. Chem.'' 1987, '''52''', 1469-1474.{{DOI|10.1021/jo00384a016}}</ref>. This stabilises the whole system, and hence endo isomer is of a lower energy than its exo counterpart. It is to be noted that secondary orbital overlap occurs on top of the HOMO-LUMO primary orbital interaction of the frontier orbitals. A simplified analogous diagram below, by Fox et al, shows the stabilising interactions with secondary orbital contribution in endo isomer as compared to the exo-isomer.

[[Image:Secondary_orb_exo.jpg|300px]] [[Image:Secondary_orb_endo.jpg|300px]]

Other than these, most of the bond lengths are similar in both structures. More significant differences appear in the newly-formed bond angle between the two components -- 92.7° in exo-TS and 96.7° in endo-TS.

In conclusion, this exercise proves that, in agreement with literature, endo product is favoured in a Diels Alder reaction, due to its transition state being more stabilised (i.e. lower in energy) than its exo isomer, hence endo product is the kinetic product while exo is thermodynamic product.<ref name="thermody dimer">W.C. Herndon, C.R. Grayson, J.M. Manion, ''J. Am. Chem. Soc..'', 2002, '''124''', 1130: {{DOI|10.1021/jo01278a003}}</ref>

=Conclusions=

In conclusions, computational methods using Gaussian calculations (with various levels of theory and different approaches) can be used to predict the viability, stability and feasibility of transition states for a pericyclic reaction, eg cycloaddition (Diels-Alder) or sigmatropic Cope rearrangement. Both steric and electronic aspects can be studied by investigating energies, geometries, structures, frontier orbitals, and reaction coordinate/reaction pathways, providing a more holistic understanding of a reaction, especially if it is kinetically controlled by studying its possible transition states.

=References=
<references />

File:ETHENE MO.LOG

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File:Secondary orb exo.jpg

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