ChemWiki - User contributions [en]

Rep:Mod3:gohsiwei

2009-11-08T14:52:36Z

Ssg07: /* Conclusion */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

In contrast to [http://en.wikipedia.org/wiki/Steric steric] reasoning which dictate that the ''anti''-conformers are more stable than the ''gauche''-conformers due to [http://en.wikipedia.org/wiki/Steric steric] repulsion between neighbouring carbon atoms in the ''gauche''-conformer, the ''gauche3''-conformer is the most stable conformer for 1,5-hexadiene.

This can be attributed to [http://en.wikipedia.org/wiki/Stereoelectronic stereoelectronic] reasons, particularly CH-π interaction. There exists favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {{DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

One drawback of utilising [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> computations to study these [http://en.wikipedia.org/wiki/Pericyclic pericyclic reactions] is that it is only capable of computing the energies of the [http://en.wikipedia.org/wiki/Transition_state transition states] and products to predict the kinetic and thermodynamic products. However, it is unable to predict whether the reaction is under kinetic or thermodynamic control, and thus it cannot determine the major product of the reaction. This can only be done experimentally.

In addition, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] may not always be capable of accurately predicting experimental observations for pericyclic reactions. Application of these methods to other known reactions must be investigated to determine its effectiveness. For example, the reactions investigated here are all thermal [http://en.wikipedia.org/wiki/Pericyclic pericyclic reactions]. The effectiveness of utilising [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> to study photoinduced pericyclic pathways was not investigated in this exercise, but a study by Gilch et al<ref name="Gilch">B. Heinz, S. Malkmus, S. Laimgruber, S. Dietrich, C. Schulz, K. Rück-Braun, M. Braun, W. Zinth, P. Gilch, ''J. Am. Chem. Soc.'', 2007, '''129''', 8577: {{DOI|10.1021/ja071396i}}</ref> was successful in using quantum mechanical calculations in supporting experimental evidence. In addition, the computational study of high-pressure induced Diels-Alder reactions was also reported by Afarinkia et al<ref name="Afarinkia">K. Afarinkia, M.J. Bearpark, A. Ndibwami, ''J. Org. Chem.'', 2003, '''68''', 7158: {{DOI|10.1021/jo0348827}}</ref>.

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-08T14:49:32Z

Ssg07: /* Conclusion */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

In contrast to [http://en.wikipedia.org/wiki/Steric steric] reasoning which dictate that the ''anti''-conformers are more stable than the ''gauche''-conformers due to [http://en.wikipedia.org/wiki/Steric steric] repulsion between neighbouring carbon atoms in the ''gauche''-conformer, the ''gauche3''-conformer is the most stable conformer for 1,5-hexadiene.

This can be attributed to [http://en.wikipedia.org/wiki/Stereoelectronic stereoelectronic] reasons, particularly CH-π interaction. There exists favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {{DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

One drawback of utilising [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> computations to study these [http://en.wikipedia.org/wiki/Pericyclic pericyclic reactions] is that it is only capable of computing the energies of the [http://en.wikipedia.org/wiki/Transition_state transition states] and products to predict the kinetic and thermodynamic products. However, it is unable to predict whether the reaction is under kinetic or thermodynamic control, and thus it cannot determine the major product of the reaction. This can only be done experimentally.

In addition, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] may not always be capable of accurately predicting experimental observations for pericyclic reactions. Application of these methods to other known reactions must be investigated to determine its effectiveness. For example, the reactions investigated here are all thermal [http://en.wikipedia.org/wiki/Pericyclic pericyclic reactions]. The effectiveness of utilising [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> to study photoinduced pericyclic pathways was not investigated in this exercise, but a study by Gilch et al<ref name="Gilch">B. Heinz, S. Malkmus, S. Laimgruber, S. Dietrich, C. Schulz, K. Rück-Braun, M. Braun, W. Zinth, P. Gilch, ''J. Am. Chem. Soc.'', 2007, '''129''', 8577: {{DOI|10.1021/ja071396i}}</ref> was successful in using quantum mechanical calculations in supporting experimental evidence. In addition, the computational study of high-pressure induced Diels-Alder reactions was also reported by Afarinkia et al<ref name="Afarinkia">K. Afarinkia, M.J. Bearpark, A. Ndibwami, ''J. Org. Chem.'', 2003, '''68''', 7158: {[DOI|10.1021/jo0348827}}</ref>.

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-08T14:43:48Z

Ssg07: /* Conclusion */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

In contrast to [http://en.wikipedia.org/wiki/Steric steric] reasoning which dictate that the ''anti''-conformers are more stable than the ''gauche''-conformers due to [http://en.wikipedia.org/wiki/Steric steric] repulsion between neighbouring carbon atoms in the ''gauche''-conformer, the ''gauche3''-conformer is the most stable conformer for 1,5-hexadiene.

This can be attributed to [http://en.wikipedia.org/wiki/Stereoelectronic stereoelectronic] reasons, particularly CH-π interaction. There exists favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {{DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental observations for pericyclic reactions. Application of these methods to other known reactions must be investigated to determine its effectiveness.

For example, the reactions investigated here are all thermal [http://en.wikipedia.org/wiki/Pericyclic pericyclic reactions]. The effectiveness of utilising [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> to study photoinduced pericyclic pathways was not investigated in this exercise, but a study by Gilch et al<ref name="Gilch">B. Heinz, S. Malkmus, S. Laimgruber, S. Dietrich, C. Schulz, K. Rück-Braun, M. Braun, W. Zinth, P. Gilch, ''J. Am. Chem. Soc.'', 2007, '''129''', 8577: {{DOI|10.1021/ja071396i}}</ref> was successful in using quantum mechanical calculations in supporting experimental evidence. In addition, the computational study of high-pressure induced Diels-Alder reactions was also reported by Afarinkia et al<ref name="Afarinkia">K. Afarinkia, M.J. Bearpark, A. Ndibwami, ''J. Org. Chem.'', 2003, '''68''', 7158: {[DOI|10.1021/jo0348827}}</ref>.

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-08T14:38:26Z

Ssg07: /* Conclusion */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

In contrast to [http://en.wikipedia.org/wiki/Steric steric] reasoning which dictate that the ''anti''-conformers are more stable than the ''gauche''-conformers due to [http://en.wikipedia.org/wiki/Steric steric] repulsion between neighbouring carbon atoms in the ''gauche''-conformer, the ''gauche3''-conformer is the most stable conformer for 1,5-hexadiene.

This can be attributed to [http://en.wikipedia.org/wiki/Stereoelectronic stereoelectronic] reasons, particularly CH-π interaction. There exists favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {{DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental observations.

For example, the reactions investigated here are all thermal [http://en.wikipedia.org/wiki/Pericyclic pericyclic reactions]. The effectiveness of utilising [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> to study photoinduced pericyclic pathways was not investigated in this exercise, but a study by Gilch et al<ref name="Gilch">B. Heinz, S. Malkmus, S. Laimgruber, S. Dietrich, C. Schulz, K. Rück-Braun, M. Braun, W. Zinth, P. Gilch, ''J. Am. Chem. Soc.'', 2007, '''129''', 8577: {{DOI|10.1021/ja071396i}}</ref> was successful in using quantum mechanical calculations in supporting experimental evidence. In addition,

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-08T12:48:48Z

Ssg07: /* Activation Energy of Reaction */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

In contrast to [http://en.wikipedia.org/wiki/Steric steric] reasoning which dictate that the ''anti''-conformers are more stable than the ''gauche''-conformers due to [http://en.wikipedia.org/wiki/Steric steric] repulsion between neighbouring carbon atoms in the ''gauche''-conformer, the ''gauche3''-conformer is the most stable conformer for 1,5-hexadiene.

This can be attributed to [http://en.wikipedia.org/wiki/Stereoelectronic stereoelectronic] reasons, particularly CH-π interaction. There exists favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {{DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-08T12:31:34Z

Ssg07: /* Geometry of TS */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

In contrast to [http://en.wikipedia.org/wiki/Steric steric] reasoning which dictate that the ''anti''-conformers are more stable than the ''gauche''-conformers due to [http://en.wikipedia.org/wiki/Steric steric] repulsion between neighbouring carbon atoms in the ''gauche''-conformer, the ''gauche3''-conformer is the most stable conformer for 1,5-hexadiene.

This can be attributed to [http://en.wikipedia.org/wiki/Stereoelectronic stereoelectronic] reasons, particularly CH-π interaction. There exists favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {{DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-08T00:10:32Z

Ssg07: /* Optimisation with HF/3-21G */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

In contrast to [http://en.wikipedia.org/wiki/Steric steric] reasoning which dictate that the ''anti''-conformers are more stable than the ''gauche''-conformers due to [http://en.wikipedia.org/wiki/Steric steric] repulsion between neighbouring carbon atoms in the ''gauche''-conformer, the ''gauche3''-conformer is the most stable conformer for 1,5-hexadiene.

This can be attributed to [http://en.wikipedia.org/wiki/Stereoelectronic stereoelectronic] reasons, particularly CH-π interaction. There exists favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {{DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-08T00:05:20Z

Ssg07: /* Activation Energy of Reaction */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

The ''gauche3''-conformer is the most stable conformer because of CH-π interaction, which is the favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {{DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-08T00:03:48Z

Ssg07: /* Optimisation with HF/3-21G */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

The ''gauche3''-conformer is the most stable conformer because of CH-π interaction, which is the favourable donation of electron density from the π<sub>C=C</sub> orbital of the C=C double bond into the σ*<sub>C-H</sub> orbital of the adjacent vinyl proton<ref name="Rocque">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T23:54:01Z

Ssg07: /* Optimisation with HF/3-21G */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

The ''gauche3''-conformer is the most stable conformer because of CH-π interaction, which is the favourable bonding interaction between the π electrons of the C=C double bond and the nearby vinyl proton<ref name="">B.G. Rocque, J.M. Gonzales, H.F. Schaefer III, ''Mol. Phys.'', 2002, '''100''', 441: {{DOI|10.1080/00268970110081412}}</ref>.

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:47:42Z

Ssg07: /* Optimisation of ''Boat'' Transition State */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' TS, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:47:04Z

Ssg07: /* Optimisation of ''Boat'' Transition State */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' TS using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:34:35Z

Ssg07: /* Conclusion */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

This exercise has shown that the elucidation of [http://en.wikipedia.org/wiki/Transition_state transition states] via the TS(Berny) optimisation method in [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/> is useful in describing the mechanism of pericyclic reactions, which are kinetically controlled.

The energy of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the activation energy of a reaction, as well as predict the selectivity of the reaction by determining the kinetic product observed. In addition, [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals] of the [http://en.wikipedia.org/wiki/Transition_state transition states] can explain the selectivity of the reaction by determining orbital overlaps which are allowed.

However, computational studies of [http://en.wikipedia.org/wiki/Transition_state transition states] are not always capable of accurately predicting experimental results.

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:27:04Z

Ssg07: /* Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile ([http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride].

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride]. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride], thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and [http://en.wikipedia.org/wiki/Maleic_anhydride maleic anhydride] is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:24:26Z

Ssg07: /* Activation Energy of Reaction */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic anhydride are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic anhydride unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic anhydride unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic anhydride unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic anhydride unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic anhydride unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic anhydride and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic anhydride.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic anhydride, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic anhydride. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic anhydride, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic anhydride subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic anhydride subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic anhydride subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:24:06Z

Ssg07: /* Molecular Orbitals of TS */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic anhydride are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic anhydride unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic anhydride unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic anhydride unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic anhydride unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic anhydride unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic anhydride and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic anhydride.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic anhydride. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic anhydride, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic anhydride subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic anhydride subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic anhydride subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:23:43Z

Ssg07: /* Geometries of TS */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic anhydride are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic anhydride unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic anhydride unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic anhydride unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic anhydride unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic anhydride unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic anhydride and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic anhydride. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic anhydride, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic anhydride subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic anhydride subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic anhydride subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:23:04Z

Ssg07: /* Molecular Orbitals of Maleic Anhydride */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic anhydride are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic anhydride. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic anhydride, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic anhydride subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic anhydride subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic anhydride subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:22:46Z

Ssg07: /* Stereoselectivity in Diels-Alder Reactions */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic anhydride. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic anhydride, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic anhydride subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic anhydride subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic anhydride subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:21:58Z

Ssg07: /* Stereoselectivity in Diels-Alder Reactions */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital<ref name="Hoffmann"/> bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:20:44Z

Ssg07: /* Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian"/>, first with the Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:19:06Z

Ssg07: /* Cope Rearrangement */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] methods, is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using first Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:18:21Z

Ssg07: /* Cope Rearrangement */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] method and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] method is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using first Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:17:48Z

Ssg07: /* Cope Rearrangement */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

In this exercise, Pople's [http://www.gaussian.com/ Gaussian]<ref name="Gaussian">Gaussian 09, Revision A.1, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009: [http://www.gaussian.com/ web]</ref>, with [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock] method and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)<ref name="DFT">JP. Hohenberg, W. Kohn, ''Phys. Rev.'', 1964, '''136''', B864: {{DOI|10.1103/PhysRev.136.B864}}</ref>] method is used to optimise the structures of molecules and [http://en.wikipedia.org/wiki/Transition_state transition states (TS)].

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using first Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:13:40Z

Ssg07: /* Further Discussion */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using first Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Conclusion =

The elucidation of transition states via the TS(Berny) optimisation method in Gaussian

<s>What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )</s>

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:11:42Z

Ssg07: /* Optimisation of TS */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using first Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

The TS(Berny) optimisation techniques was chosen over the QST2 optimisation because of the strict requirements for QST2 optimisation, in which the position and numbering of the atoms in the reactants and products must be adhered to for the optimisation to proceed successfully.

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:09:53Z

Ssg07: /* Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

In this and the next exercise, optimisation of the TS are done using first Semi-Empirical/AM1 method, followed by DFT/B3LYP/6-31G(d) method for a more accurate optimisation. Optimisation of the reagents are conducted directly with the DFT/B3LYP/6-31G(d) method as they are less computationally intensive.

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T22:06:59Z

Ssg07: /* Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T21:50:49Z

Ssg07: /* Intrinsic Reaction Coordinate */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, cyclohexene, which is shown on the right. The IRC computation can also be found at {[DOI|10042/to-2881}}.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

<s>Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained? Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').</s>

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T21:43:52Z

Ssg07: /* Intrinsic Reaction Coordinate */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, which is shown on the right.

Cyclohexene is the only product of the reaction, and has an <s>energy of ?? kJ mol<sup>-1</sup></s>. This indicates that the Diels-Alder reaction is spontaneous.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

<s>Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained? Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').</s>

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It can be found at {{DOI|10042/to-2879}}.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T21:36:02Z

Ssg07: /* Intrinsic Reaction Coordinate */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, which is shown on the right.

Cyclohexene is the only product of the reaction, and has an <s>energy of ?? kJ mol<sup>-1</sup></s>. This indicates that the Diels-Alder reaction is spontaneous.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

<s>Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained? Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').</s>

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|100px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T21:31:58Z

Ssg07: /* Intrinsic Reaction Coordinate */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, which is shown on the right.

Cyclohexene is the only product of the reaction, and has an <s>energy of ?? kJ mol<sup>-1</sup></s>. This indicates that the Diels-Alder reaction is spontaneous.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

<s>Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained? Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').</s>

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|120px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T01:19:38Z

Ssg07: /* Comparison of ''Chair'' and ''Boat'' Transition State */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup> at room temperature. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

At 0 K, the activation energy of the ''chair'' TS is 149.7 kJ mol<sup>-1</sup>, a relatively similar comparison to the experimental value of activation energy at 140±2 kJ mol<sup>-1</sup><ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>. However, the activation of the ''boat'' TS is much higher at 0 K as compared to room temperature, and at 277.6 kJ mol<sup>-1</sup>, it significantly differs from the experimental value of 187±8 kJ mol<sup>-1</sup><ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, which is shown on the right.

Cyclohexene is the only product of the reaction, and has an <s>energy of ?? kJ mol<sup>-1</sup></s>. This indicates that the Diels-Alder reaction is spontaneous.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

<s>Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained? Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').</s>

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|120px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward. It has a final energy of <s>-0.16016036 a.u.</s> <s>?? kJ mol<sup>-1</sup></s>.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It has a final energy of <s>-0.15990906 a.u.</s> <s>?? kJ mol<sup>-1</sup></s>.

By comparing the energies, the ''exo'' product is more stable than the ''endo'' product. This indicates that it should be the thermodynamic product.

However, the ''endo'' product is observed in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. This indicates that the reaction is kinetically controlled.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T01:15:00Z

Ssg07: /* Comparison of ''Chair'' and ''Boat'' Transition State */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup>. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

This is supported by comparison to literature, where the experimental value of activation energy for the ''chair'' TS is 140±2 kJ mol<sup>-1</sup> at 0 K<ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1}}</ref>, and that of the ''boat'' TS is 187±8 kJ mol<sup>-1</sup> at 0 K<ref name="Goldstein">M.J. Goldstein, M.S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''', 7147: {{DOI|10.1021/ja00775a046}}</ref>

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, which is shown on the right.

Cyclohexene is the only product of the reaction, and has an <s>energy of ?? kJ mol<sup>-1</sup></s>. This indicates that the Diels-Alder reaction is spontaneous.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

<s>Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained? Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').</s>

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|120px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward. It has a final energy of <s>-0.16016036 a.u.</s> <s>?? kJ mol<sup>-1</sup></s>.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It has a final energy of <s>-0.15990906 a.u.</s> <s>?? kJ mol<sup>-1</sup></s>.

By comparing the energies, the ''exo'' product is more stable than the ''endo'' product. This indicates that it should be the thermodynamic product.

However, the ''endo'' product is observed in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. This indicates that the reaction is kinetically controlled.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>

Rep:Mod3:gohsiwei

2009-11-07T00:49:14Z

Ssg07: /* Comparison of ''Chair'' and ''Boat'' Transition State */

= Cope Rearrangement =

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangment]<ref name="Cope">A.C. Cope, E.M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''', 441: {{DOI|10.1021/ja01859a055}}</ref> reaction is a [3,3]-[http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangment] reaction.

As with other [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement sigmatropic rearrangments], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which one σ bond is formed while another is broken simultaneoulsy in a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

1,5-hexadiene undergoes the reaction shown below.

[[Image:Copey.gif]]

== Optimisation of 1,5-hexadiene Conformers ==

=== Optimisation with HF/3-21G ===

Two ''anti''- and one ''gauche''- conformer of 1,5-hexadiene were optimised using the HF/3-21G method. They have been identified using the table in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] from Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

{| class="wikitable" border="1"
|+ ''Table 1: Optimisation of 1,5-hexadiene Conformers''
! Property !! ''Anti1''-Conformer !! ''Anti2''-Conformer !! ''Gauche3''-Conformer
|-
| Final Energy / a.u. ||align="center"| -231.69260 ||align="center"| -231.69254 ||align="center"| -231.69266
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608308.9 ||align="center"| -608308.7||align="center"| -608309.1
|-
| Point Group ||align="center"| C<sub>2</sub> ||align="center"| C<sub>i</sub> ||align="center"| C<sub>1</sub>
|-
| Molecule ||align="center"|<jmol><jmolApplet><title>anti1</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti1.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>anti2</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_anti2.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|<jmol><jmolApplet><title>gauche3</title><color>black</color><size>150</size><script>cpk -20;</script><uploadedFileContents>Cope_gauche3.mol</uploadedFileContents></jmolApplet></jmol>
|}

<s>The ''gauche3''-conformer is the most stable conformer because</s>

=== Optimisation of ''Anti2''-Conformer with DFT/B3LYP/6-31G(d) ===

==== Comparison between HF/3-21G and DFT/B3LYP/6-31G(d) Optimisation ====

The HF/3-21G optimised ''anti2''-conformer was further optimised with the DFT/B3LYP/6-31G(d) method to yield an improved structure with lower energy.

{| class="wikitable" border="1"
|+ ''Table 2: Optimisation of Anti2-1,5-hexadiene''
! Properties !! DFT/B3LYP/6-31G(d) Method !! HF/3-21G Method
|-
| Final Energy / a.u. ||align="center"| -234.61170 ||align="center"| -231.69254
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615973.0 ||align="center"| -608308.7
|-
| C1-C4 dihedral angle / ° ||align="center"| 118.5 ||align="center"| 114.7
|}

While there was little change in the bond lengths and bond angles between the DFT/B3LYP/6-31G(d) and HF/3-21G optimised structures, the dihedral angle between C1 and C4 increased significantly from 114.7° to 118.5°.

==== Comparison of Energies at 0K and 298K ====

The energy of ''anti2''-1,5-hexadiene can be resolved into potential and kinetic energies, and from there the enthalphy and free energy of dissociation can be calculated. As these parameters are temperature-dependent, the computation was repeated at 0 K and 298.15 K to study the effect of temperature on the energies of ''anti2''-1,5-hexadiene.

{| class="wikitable" border="1"
|+ ''Table 3: Temperature-Dependence of Anti2-1,5-hexadiene Energies''
! Energies !! At 0 K !! At 298.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.61170
|-
| Sum of Electronic and Zero-point Energies) / a.u. ||align="center"| -234.46877 ||align="center"| -234.46921
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.46143 ||align="center"| -234.46186
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.46049 ||align="center"| -234.46091
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.50037 ||align="center"| -234.50082
|-
| Link ||align="center"| {{DOI|10042/to-2706}} ||align="center"| {{DOI|10042/to-2705}}
|}

The total energy of the molecule becomes more negative as temperature increases. This is expected since the molecule absorbs more heat at higher temperatures, and thus thermal contribution to the energies increase.

== Optimisation of Transition State Structures ==

=== Optimisation of ''Chair'' Transition State ===

==== Comparison of TS Optimisation Methods ====

The ''chair'' TS was optimised using HF/3-21G via two types of optimisation.

''Opt 1'': A direct TS(Berny) Optimisation

''Opt 2'': A ModRedundant Minimisation with frozen C-C lengths, at 2.20 Å, for the C atoms involved in bond breaking/forming; followed by a TS(Berny) Optimisation with Hessian Derivative calculation for the C-C bond breaking/forming lengths

The table below shows a comparison between the two techniques:

{| class="wikitable" border="1"
|+ ''Table 4: Optimisation of Chair TS''
! Properties !! ''Opt 1'' !! ''Opt 2''
|-
| Final Energy / a.u. ||align="center"| -234.61933 ||align="center"| -231.61932
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -615993.0 ||align="center"| -608116.5
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -818 ||align="center"| -818
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.02 ||align="center"| 2.02
|}

[[Image:Tschair818vib.jpg|150px|right]]As expected, both techniquess gave the same results for the energy of the TS. In addition, both techniques report the same C-C bond breaking/forming length, at 2.02 Å. They also give an imaginary frequency of -818 cm<sup>-1</sup>, which upon animation of the vibration, shows asynchronous bond formation i.e. one bond breaks and the other forms simultaneously, as shown on the right. This is expected for the Cope Rearrangment, since it is an electrocyclic reaction.

In fact, the extra step of ModRedundant Minimisation in ''Opt 2'' is unnecessary for this optimisation, since the guess structure for the ''chair'' TS is similar to the optimised TS.

However, for molecules in which the guess structure differs greatly from the true conformation, a direct TS(Berny) optimisation, as in ''Opt 1'', cannot optimise the guess structure. This is observed later in the optimisation of the TS for the Diels-Alder reaction. In such situations, ''Opt 2'' is applied. The ModRedundant Minimisation is used to first optimise the separate units of the TS individually, which the bond breaking/forming distance frozen. Upon completion, the coordinates are unfrozen, and the TS(Berny) optimisation can then be used to optimise the bond breaking/forming distance as part of the entire TS optimisation.

==== Intrinsic Reaction Coordinate ====

The Intrinsic Reaction Coordinate (IRC) calculation was conducted on the HF/3-21G optimised ''chair'' TS. The purpose of this calculation is to model the transformation from the ''chair'' TS down the potential energy surface to the final product of the reaction.

Initially, the IRC calculation was done with 50 points on the IRC pathway and computation of the force constants only once. The output showed no bond formation process, and hence it was deduced that the calculation was incomplete, as a 1,5-hexadiene structure was expected. Analysis of the IRC pathway also showed that the minimum on the potential energy surface has not been reached, as the points on the pathway do not reach an asymptote.

Th calculation was repeated with the same number of points on the IRC pathway, but this time with computation of force constants at every step. This allows the IRC calculation to determine if it is moving along the minimum energy pathway down the potential energy surface, and make changes to the pathway when necessary. This time, the process of bond forming to arrive at 1,5-hexadiene was observed in the output, and the IRC pathway also reached a plateau with minimum energy.

{| class="wikitable" border="1"
|+ ''Table 2: Comparison of IRC Computations''
! Properties !! Force Constant Computation Once !! Force Constant Computation Always
|-
| Final Energy / a.u. ||align="center"| -231.68815 ||align="center"| -231.69167
|-
| IRC Pathway ||align="center"|[[Image:Tschairbadircpath.jpg|200px]]||align="center"|[[Image:Ircpath.jpg|200px]]
|-
| Structure ||align="center"|[[Image:Tschairbadirc.jpg|160px]]
{{DOI|10042/to-2845}}
|align="center"|[[Image:Irc.jpg|200px]]
{{DOI|10042/to-2702}}
|}

Analysis of the product in the successful IRC, with force constant calculation always on, showed that it is the ''gauche2''-1,5-hexadiene conformer. The minimum energy reached in the IRC pathway, at -231.69167 a.u., also corresponds to the energy of the ''gauche2'' conformer in [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1] of Dr. Mike Bearpark's 3rd Year Computational Chemistry Lab [http://www.ch.ic.ac.uk/wiki/index.php/Mod:phys3 Module 3] course wiki.

=== Optimisation of ''Boat'' Transition State ===

The ''boat'' TS was optimised via QST2 Optimisation, which requires the input of both the reagent and product molecules, and numbering of the atoms in the molecule to identify their positions in both reagent and product.

The following table shows the successful optimisation of the ''boat'' transition state using QST2 Optimisation.

{| class="wikitable" border="1"
|+ ''Table 5: Optimisation of Boat TS''
! Properties !! ''QST2''
|-
| Final Energy / a.u. ||align="center"| -231.60282
|-
| Final Energy / kJ mol<sup>-1</sup> ||align="center"| -608073.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -840
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 2.14
|}

[[Image:Boatvib.jpg|120px|right]]The presence of the imaginary frequency at -840 cm<sup>-1</sup> indicates that a TS was found. The movement of the bonds upon animation is shown in the diagram on the right. The vibration shows asynchronous bond formation ie. one bond is formed while the other is broken, as expected in the Cope Rearrangement.

The QST2 Optimisation is more tedious than the TS(Berny) optimisation utilised for the ''chair'' transition state, as the exact geometry and numbering of the structures must be achieved for the optimisation to work. In practice, the QST2 optimisation failed to work even when the correct numbering and geometry was achieved; and the input had to be recreated before the calculation could succeed.

Hence, in further TS optimisations, the TS(Berny) Optimisation will be utilised over the QST2 method for its ease of usage.

=== Comparison of ''Chair'' and ''Boat'' Transition State ===

To obtain more accurate descriptions of both TS, the HF/3-21G optimised structures were sent for a final TS(Berny) optimisation and frequency calculation using the DFT/B3LYP/6-31G(d) method.

For the ''chair'' TS, all additional selections and keywords were retained, while for the ''boat'' TS, the additional keyword "Opt=CalcFC" had to be added. This allows the force constants to be calculated for the calculation to succeed.

A comparison of the structure and energies of the ''chair'' and ''boat'' TS, both at 0 K and 298.15 K, is made below.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Chair and Boat TS''
! Properties !! ''Chair'' TS at 0 K !! ''Chair'' TS at 298.15 K !! ''Boat'' TS at 0 K !! ''Boat'' TS at 295.15 K
|-
| Electronic Energy / a.u. ||align="center"| -234.55698 ||align="center"| -234.55698 ||align="center"| -234.54309 ||align="center"| -234.54309
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| -234.41449 ||align="center"| -234.41493 ||align="center"| -234.40190||align="center"| -234.40234
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| -234.40858 ||align="center"| -234.40901 ||align="center"| -234.39558||align="center"| -234.39601
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| -234.40763 ||align="center"| -234.40807 ||align="center"| -234.39464||align="center"| -234.39506
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| -234.44336 ||align="center"| -234.44382||align="center"| -234.43130||align="center"| -234.43175
|-
| Activation Energy / a.u. ||align="center"| 0.05701 ||align="center"| 0.05655 ||align="center"| 0.10573 ||align="center"| 0.06862
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 149.7 ||align="center"| 148.5 ||align="center"| 277.6||align="center"| 180.2
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -566 ||align="center"| -566 ||align="center"| -531||align="center"| -531
|-
| C-C Bond Breaking/Forming Length / Å ||align="center"| 1.97 ||align="center"| 1.97 ||align="center"| 2.21||align="center"| 2.21
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>chair</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tschair.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2874}} for 0 K, and {{DOI|10042/to-2703}} for 298.15 K
|align="center"|<jmol><jmolApplet><title>Tsboat</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Tsboatb3lyp.mol</uploadedFileContents></jmolApplet></jmol>||The jobs can be found at: {{DOI|10042/to-2704}} for 0 K, and {{DOI|10042/to-2867}}for 298.15 K
|}

''Note: The activation energies of the TS are measured with respect to anti2-1,3-hexadiene. Activation energy can be calculated by subtracting the electronic energy of anti2-1,3-hexadiene, at -234.50037 a.u., from that of the TS.''

Comparing the results from the optimisation of the TS using DFT/B3LYP/6-31G(d) with those done with HF/6-31G, it is observed that the energies of both TS have decreased significantly, as have the imaginary frequencies. The C-C bond breaking/forming length has also changed.

This is because DFT/B3LYP/6-31G(d) is a higher level optimisation than HF/6-31G, and it adjusts the structure of TS, such as the C-C bond forming/breaking length, to further decrease its energy. The decrease in imaginary frequency indicates a gentler gradient of the potential energy curve at the maximum, indicating that this is a better optimisation.

From the table, it is observed that the ''chair'' TS is 0.01207 a.u., or 31.7 kJ mol<sup>-1</sup>, more stable than the ''boat'' TS. This results in the ''chair'' TS having a lower activation energy of 148.5 kJ mol<sup>-1</sup> compared to the ''boat'' TS, at 180.2 kJ mol<sup>-1</sup>. Hence, the ''chair'' TS is the expected TS for the Cope rearrangement of 1,5-hexadiene.

This is supported by comparison to literature, where the experimental value of activation energy for the ''chair'' TS is 140±2 kJ mol<sup>-1</sup> at 0 K<ref name="Doering">W.vonE. Doering, V.G. Toscano, G.H. Beasley, ''Tetrahedron'', 1971, '''27''', 5299: {{DOI|10.1016/S0040-4020(01)91694-1 }}</ref>.

= Diels-Alder Cycloaddition of ''Cis''-Butadiene and Ethene =

The [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer">J. Sauer, R. Sustmann, ''Angew. Chem'', 1980, '''19''', 779: {{DOI|10.1002/anie.198007791}}</ref> reaction is a [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition] reaction between a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] diene system and an alkene system to form a cyclohexene system. It is otherwise known as a [4+2] or п<sub>4s</sub>+п<sub>4s</sub> [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition].

As with other [http://en.wikipedia.org/wiki/Cycloaddition cycloaddition reactions], the reaction occurs via a single cyclic, concerted [http://en.wikipedia.org/wiki/Transition_state transition state] in which two σ bonds are formed between the termini of the two [http://en.wikipedia.org/wiki/Conjugated_system conjugated] п systems.

''Cis''-butadiene and ethene react in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction shown below.

[[Image:DA1.gif]]

== Optimisation of Reagents ==

=== Optimisation of ''Cis''-Butadiene ===

==== Energy of ''Cis''-Butadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cis-Butadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -155.98649
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -155.90114
|-
| Sum of Electronic and Thermal Energies||align="center"| -155.89644
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -155.89549
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -155.92785
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cis-butadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of ''Cis''-Butadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ''cis''-butadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.34455 a.u.
|align="center"| 0.01795 a.u.
|-
| [[Image:Cis-butadieneHOMO.jpg|160px]]
| [[Image:Cis-butadieneLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO (highest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO]) of ''cis''-butadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, as the phases are inverted.

Conversely, the LUMO (lowest occupied [http://en.wikipedia.org/wiki/Molecular_orbital MO])is symmetric (S) with respect to the plane, since it can be reflected in the plane of symmetry of the molecule.

These assignments of orbital symmetry are important in [http://en.wikipedia.org/wiki/Pericyclic pericyclic] reactions, as they determine whether the reaction will proceed, and also the [http://en.wikipedia.org/wiki/Stereochemistry stereochemical] outcome of the reaction.

=== Optimisation of Ethene ===

==== Energy of Ethene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Ethene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -78.58746
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -78.53623
|-
| Sum of Electronic and Thermal Energies||align="center"| -78.53319
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -78.53225
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -78.55776
|}

==== Molecular Orbitals of Ethene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of ethene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.26664 a.u.
|align="center"| 0.01881 a.u.
|-
| [[Image:AlkeneHOMO.jpg|150px]]
| [[Image:AlkeneLUMO.jpg|170px]]
|-
|align="center"| Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (S) wrt Plane
|}

The HOMO of ethene is symmetric (S) with respect to the plane of symmetry, while the LUMO is anti-symmetric (AS).

This is opposite to the symmetries of the HOMO and LUMO in ''cis''-butadiene.

Thus, by the conservation of orbital symmetry<ref name="orbsymm">R. Hoffmann, R.B. Woodward, ''Acc. Chem. Res.'', 1968, '''1''', 17: {{DOI|10.1021/ar50001a003}}</ref>, which states that only orbitals of the same symmetry can interact, orbital interaction between the HOMO of ''cis''-butadiene and the LUMO of ethene, as well as between the LUMO of ''cis''-butadiene and the HOMO of ethene, can occur.

== Optimisation of TS ==

The optimisation of the TS was conducted in three consecutive steps:
# Optimisation to a Minimum using Semi-empirical/AM1 method with ModRedundant to freeze the C-C bond forming distance at 2.1 Å
# Optimisation to a TS(Berny)using Semi-empirical/AM1 method with ModRedundant to calculate the derivative of the C-C bond forming distance, which was unfrozen
# Optimisation and Frequency calculation to a TS(Berny) using DFT/B3LYP/6-31G(d) method with the same techniques as in Step 2 but with additional keyword: Opt=NoEigen

=== Energy of TS ===

The following results were obtained from the Optimisation and Frequency TS(Berny) calculations using Semi-empirical/AM1 and DFT/B3LYP/6-31G(d).

{| class="wikitable" border="1"
|+ ''Table 5: Energy of TS''
! Properties !! Semi-empirical/AM1 !! DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u.||align="center"| 0.121165 ||align="center"| -234.54390
|-
| Sum of Electronic and Zero-Point Energies / a.u.||align="center"| 0.25328 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Energies / a.u.||align="center"| 0.25945 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Enthalpies / a.u.||align="center"| 0.26040 ||align="center"| -234.54390
|-
| Sum of Electronic and Thermal Free Energies / a.u.||align="center"| 0.22402 ||align="center"| -234.54390
|-
| Imaginary Frequency / cm<sup>-1</sup>||align="center"| -956 ||align="center"| -525
|-
| C-C Bond Forming Length / Å ||align="center"| 2.12 ||align="center"| 2.27
|-
| Structure ||align="center"|<jmol><jmolApplet><title>DA1</title><color>black</color>
<size>150</size><script>cpk -20</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|{{DOI|10042/to-2701}}
|}

In both optimisations, the presence of one imaginary frequency indicates that a TS has been found.

Comparing the results from the two optimisations, there is a vast difference in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resultign in a lower total energy of the TS.

=== Geometry of TS ===

{|
|The Jmol Applet on the right shows the indicates the bond lengths and bond angles in the TS. Since the TS is symmetrical, only one of each pair of similar bonds/angles are shown. Also, the dihedral angle between C1 and C4 of ''cis''-butadiene, which is 0.0°, is not shown.

The bond breaking/forming length in the TS is 2.27 Å from the DFT/B3LYP/6-31G method.

In contrast, the typical C''sp<sup>3</sup>''-C''sp<sup>3</sup>'' bond length is 1.53 Å, and the typical unconjugated C''sp<sup>2</sup>''-C''sp<sup>2</sup>'' bond length is 1.48 Å<ref name="BL">F.H. Allen, O. Kennard, D.G. Watson, L. Brammer, A.G. Orphen, R. Taylor, ''J. Chem. Soc., Perkin Trans. 2'', '''1987''', S1-S19: {{DOI|10.1039/P298700000S1}}</ref>. However, the bond forming length is shorter than twice the van der Waals radius of the C atom, which is 1.70 Å for aliphatic carbons<ref name="Bondi">A. Bondi, ''J. Phys. Chem.'', 1964, '''68''', 441: {{DOI|10.1021/j100785a001}}</ref>.

This indicates that the C atoms are approaching within their van der Waals radii for bond formation, but the bond has not yet been fully formed, and so the distance between the atoms is still larger than that of a C-C single bond.
| <jmol><jmolApplet><title>DA1</title><color>black</color>
<size>200</size><script>zoom 150; cpk -20; measure 1 2; measure 1 13; measure 13 8; measure 5 8; measure 1 2 11; measure 2 11 5; measure 11 5 8</script><uploadedFileContents>DA1.mol</uploadedFileContents></jmolApplet></jmol>
|}

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.32393 a.u.
|align="center"| 0.02315 a.u.
|-
| [[Image:DA1_TS_HOMO.jpg|165px]]
| [[Image:DA1_TS_LUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt plane
|align="center"| Symmetric (S) wrt plane
|}

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene. Meanwhile, the LUMO of the TS is built up of the LUMO of ''cis''-butadiene and the HOMO of ethene.

This is consistent with the conservation of orbital symmetry<ref name="orbsymm"/>, which states that the symmetry of the orbital formed form the overlap of two orbitals, must be the same as that of the initial orbitals. It also fits with MO theory since only filled orbitals can overlap with empty orbitals and vice versa.

Comparing the energies of the [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the reagents, it is also evident that the overlap of the HOMO of ''cis''-butadiene and the LUMO of ethene will form a lower energy orbital than the overlap of the LUMO of ''cis''-butadiene and the HOMO of ethene. This is observed in the HOMO and LUMO of the TS.

== Activation Energy of Reaction ==

Summing the free energies of ''cis''-butadiene and ethene, the total free energy of the reagents is -234.48561 a.u. In comparison, the free energy of the TS is -234.43289 a.u.

Hence, the activation energy, which is the difference in energy between the TS and the reagents, is 0.052720 a.u., or 138.4 kJ mol<sup>-1</sup>.

This is supported by experimental data<ref name="Guner">V. Guner, K.S. Khuong, A.G. Leach,, P.S. Lee, M.D. Bartberger, K.N. Houk, ''J. Phys. Chem. A, 2003, '''107''', 11445: {[DOI|10.1021/jp035501w}}</ref> from literature which quotes the activation energy to be 115 ± 8 kJ mol<sup>-1</sup> at 0 K.

== From TS to Product ==

=== Vibrational Frequencies ===

As discussed above, the imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

[[Image:DA1vib.JPG|120px|left]][[Image:DA1vib+147.jpg|120px|right]]The frozen animation of the imaginary vibration of the TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions, which lead forward to the products. The formation of the two bonds is synchronous, as expected for the concerted [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction.

The lowest positive frequency vibration at 147 cm<sup>-1</sup> is a rock, shown on the right. In contrast to the imaginary vibration, the motion of the two termini of each unit of the TS is asynchronous ie. one end of the ''cis''-butadiene and ethene termini move towards each other, while at the other end they move away from each other. Also, the magnitude of vibration is smaller than that of the imaginary vibration.

=== Intrinsic Reaction Coordinate ===

[[Image:Da1_irc.jpg|120px|right]]IRC was utilised to track the reaction from the TS down the potential energy surface via the minimum energy route to the product, which is shown on the right.

Cyclohexene is the only product of the reaction, and has an <s>energy of ?? kJ mol<sup>-1</sup></s>. This indicates that the Diels-Alder reaction is spontaneous.

== Conservation of Orbital Symmetry ==

By conservation of orbital symmetry<ref name="orbsymm"/>, the transition of [http://en.wikipedia.org/wiki/Molecular_orbital MOs] from those in reagents to those in products via a concerted reaction must not change the symmetry of the orbitals. Reactions which follow these rule are symmetry-allowed, while those which do not are forbidden.

In the Diels-Alder reaction of ''cis''-butadiene with ethene, orbital overlap between the anti-symmetric (AS) HOMO of ''cis''-butadiene and the AS LUMO of ethene forms the HOMO of the cyclohexene product, which is also AS with respect to the plane of symmetry of the molecule. This retention of symmetry can also be observed in the HOMO of the TS.

Another rule in the conservation of orbital symmetry<ref name="orbsymm"/> is that only orbitals of different symmetry can cross in energy. This rule is also applied since the LUMO of ''cis''-butadiene, which is symmetric (S) with respect to the plane, overlaps with the S HOMO of ethene to form the LUMO of the TS and thus product, which retains the symmetry.

Energetically, the reaction is also allowed because the energy gap between the HOMO of ''cis''-butadiene and the LUMO of ethene is small, and the product is more stable than the reagents. In addition, the small activation energy makes this reaction kinetically favourable as well.

= Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride =

As with ''cis''-butadiene and ethene, the reaction between cyclohexa-1,3-diene and maleic anhydride is also a [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. However, in contrast to the above reaction, two products are formed in this reaction, as shown below.

[[Image:DA2.gif]]

The ''endo'' and ''exo'' products are [http://en.wikipedia.org/wiki/Diastereoselectivity diastereomers], and their formation is dependent on the direction of attack of the diene (cyclohexa-1,3-diene) on the dienophile (maleic anhydride). As with most [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reactions, the ''endo'' isomer is the major product of this reaction, which will be explained in the following exercise.

<s>Comment on the structural difference between the endo and exo form. Why do you think that the exo form could be more strained? Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system. What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').</s>

== Optimisation of Reactants ==

=== Optimisation of Cyclo-1,3-hexadiene ===

==== Energy of Cyclo-1,3-hexadiene ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Cyclo-1,3-hexadiene''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -233.41892
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -233.29610
|-
| Sum of Electronic and Thermal Energies||align="center"| -233.29092
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -233.28998
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -233.32370
|-
| Structure ||align="center"|<jmol><jmolApplet><title>Cyclohexadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Cyclohexadiene.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Cyclo-1,3-hexadiene ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of cyclo-1,3-hexadiene are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.01711 a.u.
|align="center"| -0.20551 a.u.
|-
| [[Image:CyclohexadienHOMO.jpg|160px]]
| [[Image:CyclohexadienLUMO.jpg|170px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Symmetric (S) wrt Plane
|}

The HOMO of cyclo-1,3-hexadiene is anti-symmetric (AS) with respect to the reflection plane of the molecule, while the LUMO is symmetric (S) with respect to the plane. This is similar to the symmetries of the HOMO and LUMO for ''cis''-butadiene.

The DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive. However, this is a small negative energy and can be attributed to inadequacies in the method.

=== Optimisation of Maleic Anhydride ===

==== Energy of Maleic Anhydride ====

{| class="wikitable" border="1"
|+ ''Table 5: Energy of Maleic Anhydride''
! Energy Type !! Energy / a.u.
|-
| Electronic Energy||align="center"| -379.28954
|-
| Sum of Electronic and Zero-Point Energies||align="center"| -379.23366
|-
| Sum of Electronic and Thermal Energies||align="center"| -379.22847
|-
| Sum of Electronic and Thermal Enthalpies||align="center"| -379.22753
|-
| Sum of Electronic and Thermal Free Energies||align="center"| -379.26273
|-
| Structure ||align="center"|<jmol><jmolApplet><title>cisbutadiene</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>Maleic_anhydride.mol</uploadedFileContents></jmolApplet></jmol>
|}

==== Molecular Orbitals of Maleic Anhydride ====

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of maleic anhydride are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|
!HOMO
!LUMO
|-
|align="center"| -0.11714 a.u.
|align="center"| -0.29929 a.u.
|-
| [[Image:MaleicHOMO.jpg|200px]]
| [[Image:MaleicLUMO.jpg|160px]]
|-
|align="center"| Anti-Symmetric (AS) wrt Plane
|align="center"| Anti-Symmetric (AS) wrt Plane
|}

Both the HOMO and the LUMO of maleic acid are anti-symmetric (AS) with respect to the plane of symmetry (reflection) the molecule, as the phases are inverted.

Again, DFT/B3LYP/6-31G(d) calculation shows the energy of the LUMO to be negative, when it is expected to be positive.

== Optimisation of TS ==

The optimisation of the TS was conducted in the same manner in the optimisation of the TS of the Diels-Alder reaction between ''cis''-butadiene and ethene above, using the 3-step method.

=== Energies of TS ===

The following table shows the results for both the Semi-empirical/AM1 and DFT/B3LYP/6-31G(d) optimisations of the TS.

{| class="wikitable" border="1"
|+ ''Table 5: Comparison of Endo and Exo TS''
! Properties !! ''Endo'' TS with Semi-empirical/AM1 !! ''Endo'' TS with DFT/B3LYP/6-31G(d) !! ''Exo'' TS with Semi-empirical/AM1 !! ''Exo'' TS with DFT/B3LYP/6-31G(d)
|-
| Electronic Energy / a.u. ||align="center"| -0.05151 ||align="center"| -612.68340 ||align="center"| -0.05042 ||align="center"| -612.67931
|-
| Sum of Electronic and Zero-Point Energies / a.u. ||align="center"| 0.13349 ||align="center"| -612.50214 ||align="center"| 0.13488 ||align="center"| -612.49801
|-
| Sum of Electronic and Thermal Energies / a.u. ||align="center"| 0.14368 ||align="center"| -612.49179 ||align="center"| 0.14488 ||align="center"| -612.48766
|-
| Sum of Electronic and Thermal Enthalpies / a.u. ||align="center"| 0.14463 ||align="center"| -612.49084 ||align="center"| 0.14583 ||align="center"| -612.48672
|-
| Sum of Electronic and Thermal Free Energies / a.u. ||align="center"| 0.09735 ||align="center"| -612.53833 ||align="center"| 0.09912 ||align="center"| -612.53427
|-
| Imaginary Frequency / cm<sup>-1</sup> ||align="center"| -806 ||align="center"| -447 ||align="center"| -812 ||align="center"| -448
|-
| C-C Bond Forming Length / Å ||align="center"| 2.17 ||align="center"| 2.27 ||align="center"| 2.17||align="center"| 2.29
|-
| TS Structure ||align="center"|<jmol><jmolApplet><title>endo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2700}}
|align="center"|<jmol><jmolApplet><title>exo</title><color>black</color>
<size>150</size><script>cpk -20;</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>||align="center"|{{DOI|10042/to-2699}}
|}

The presence of one imaginary frequency indicates that the optimised structure is indeed a TS.

Comparing the results from the two optimisations, there is a vast differnce in energies. However, there is no need for alarm at this difference since the semi-empirical and DFT methods report energies differently, and hence they cannot be compared. The increase in C-C bond forming length, however, can be attributed to the greater accuracy of the DFT/B3LYP/6-31G(d) optimisation. The increase in bond length results in a less steep potential energy slope to the TS maximum, as indicated by the decrease in imaginary frequency, and also resulting in a lower total energy of the TS.

From the table, it is observed that the ''endo'' TS is 0.00406 a.u., or 10.7 kJ mol<sup>-1</sup>, more stable than the ''exo'' TS. The difference in energies of the ''endo'' and ''exo'' TS can be attributed to secondary orbital interactions in the ''endo'' TS which stabilises the structure, and will be discussed in the section on molecular orbitals.

This results in the ''endo'' TS having a lower activation energy to the ''exo'' TS as shown in the section on activation energies below. Hence, the ''endo'' TS is is predicted to give the kinetic product of the reaction.

=== Geometries of TS ===

A comparison of the bond lengths and bond angles of the ''endo'' and ''exo'' TS are made on the below.

The most evident difference between the two TS is that in the ''endo'' isomer, the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward; while in the ''exo'' isomer the H atoms on the maleic acid unit are pointing downward and the 5-membered ring is pointing upward.

{|
! ''Endo'' TS !! ''Exo'' TS
|-
| <jmol><jmolApplet><title>endo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_endo.mol</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet><title>exo</title><color>black</color>
<size>300</size><script>zoom 150; cpk -20; measure 1 4; measure 2 17; measure 1 2; measure 2 7; measure 7 10; measure 17 21; measure 21 15; measure 21 23; measure 16 17; measure 1 4 3; measure 4 3 10; measure 10 3 16; measure 3 16 20; measure 16 20 15; measure 20 15 21; measure 20 16 17; measure 3 10 7; measure 4 20; measure 1 2 17 19</script><uploadedFileContents>DA2_exo.mol</uploadedFileContents></jmolApplet></jmol>
|}

There is little difference between the bond lengths and bond angles not involved in the C-C bond formation. Also, even the C-C bond forming length is similar for the TS, at 2.27 Å in the ''endo'', and at 2.29 Å in the ''exo''.

However, there is remarkable difference between bond angles and especially dihedral angles at the C-C bond forming site. The bond angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the bond forming C on the maleic acid unit is 94.3° for the ''endo'' TS, and 99.1° in the ''exo''. Also, the dihedral angle between the C=C double bond on the cyclo-1,3-hexadiene unit and the C-H bond on the maleic acid unit is 175.1° for the ''endo'' TS, and 70.6° in the ''exo''.

Another difference which cannot be overlooked is the distance between the C of the carbonyl group in the maleic acid unit and the non-reacting C of the C=C double bond on theh cyclo-1,3-hexadiene unit nearest to it. In the ''endo'' TS, this distance is 2.99 Å, while in the ''exo'' TS it is 3.88 Å.

This distance is important for the secondary orbital overlap between the π*<sub>C=O</sub> orbital from maleic acid and the π<sub>C-C</sub> orbital from cyclo-1,3-hexadiene, which stabilises the TS. The shorter distance in the ''endo'' TS indicates that secondary orbital overlap can occur, while the longer distance in the ''exo'' TS spells the absence of this overlap.

=== Molecular Orbitals of TS ===

The reactive [http://en.wikipedia.org/wiki/Molecular_orbital molecular orbitals (MOs)] of both the ''endo'' and ''exo'' TS are shown below, with their symmetries with respect to (wrt) the plane of symmetry (reflection) of the molecule indicated.

{|border="1"
! colspan="2"| ''Endo'' TS
! colspan="2"| ''Exo'' TS
|-
|align="center"| '''HOMO''' (-0.24230 a.u.)
|align="center"| '''LUMO''' (-0.06771 a.u.)
|align="center"| '''HOMO''' (-0.07840 a.u.)
|align="center"| '''LUMO''' (-0.24215 a.u.)
|-
| [[Image:DA2_endo_HOMO.jpg|170px]]
| [[Image:DA2_endo_LUMO.jpg|170px]]
| [[Image:DA2_exo_HOMO.jpg|170px]]
| [[Image:DA2_exo_LUMO.jpg|170px]]
|-
|align="center"| Anti-symmetric (AS) wrt plane
|align="center"| AS
|align="center"| AS
|align="center"| AS
|}

Both the HOMO and the LUMO of the ''endo''and ''exo'' TS are antisymmetric with respect to the plane of symmetry of the TS. Again, DFT/B3LYP/6-31G(d) calculation shows the energies of the LUMOs to be negative, when it is expected to be positive.

By comparing the reactive [http://en.wikipedia.org/wiki/Molecular_orbital MOs] of the TS with those of the reagents, it is observed that the HOMO of the TS is derived from the overlap of the HOMO of cyclo-1,3-hexadiene and the LUMO of maleic acid.

== Activation Energy of Reaction ==

Summing the free energies of cyclo-1,3-hexadiene and maleic acid, the total free energy of the reagents is -612.58643 a.u.

This is utilised to calculate the activation energy for the ''endo'' and ''exo'' TS, as shown below.

{| class="wikitable" border="1"
|+ ''Table 10: Activation Energies for TS''
! Activation Energy !! ''Endo'' TS !! ''Exo'' TS
|-
| Activation Energy / a.u. ||align="center"| 0.04810 ||align="center"| 0.05216
|-
| Activation Energy / kJ mol<sup>-1</sup> ||align="center"| 126.3 ||align="center"| 136.9
|}

''Note: The activation energies of the TS are measured with respect to the total free energy of the reagents. Activation energy can be calculated by subtracting the total free energy of the reagents, at -612.58643 a.u., from that of the TS.''

<s>compare to experimental</s>

From the table, it is observed that the activation energy for the ''endo'' TS is lower than that of the ''exo'' TS by 10.6 kJ mol<sup>-1</sup>. Hence, the ''endo'' product is expected to be the kinetic product of the reaction.

== From TS to Product ==

=== Vibrational Frequencies ===

[[Image:Da2endovib.JPG|120px|left]][[Image:Da2exovib812.jpg|120px|right]]The imaginary vibrations of the TS indicate bond breaking/forming interactions in the TS, which either lead forward to the reagents or backward to the products.

The frozen animation of the imaginary vibration of the ''endo'' TS is shown on the left. The blue arrows indicate the bond breaking interactions, which lead backward to the reagents, while the black arrows indicate the bond forming interactions which form the products.

The frozen animation of the imaginary vibration of the ''exo'' TS is shown on the right, this time with the blue arrows indicating bond forming interactions to form the products.

In both the ''endo'' and the ''exo'' TS, the bond formation is synchronous ie. both bonds are formed concurrently, as expected for cycloadditions, which are concerted reactions.

=== Intrinsic Reaction Coordinate ===

[[Image:Da2endo_IRC.jpg|120px|left]][[Image:Da2exo_irc.jpg|140px|right]]IRC was utilised to track the reaction from the Semi-empirical/AM1 optimised TS down the potential energy surface to the product for both the ''endo'' and ''exo'' TS.

The ''endo'' product is shown on the left, and can be found at {{DOI|10042/to-2847}}. It can be identified by the H atoms on the 5-membered ring pointing upward, and the ring is pointing downward. It has a final energy of <s>-0.16016036 a.u.</s> <s>?? kJ mol<sup>-1</sup></s>.

For the ''exo'' product, illustrated on the right, the H atoms on the 5-membered ring are pointing downward, and the ring is pointing upward. It has a final energy of <s>-0.15990906 a.u.</s> <s>?? kJ mol<sup>-1</sup></s>.

By comparing the energies, the ''exo'' product is more stable than the ''endo'' product. This indicates that it should be the thermodynamic product.

However, the ''endo'' product is observed in the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction. This indicates that the reaction is kinetically controlled.

== Stereoselectivity in Diels-Alder Reactions ==

The selection of the ''endo'' product follows Alder's [http://en.wikipedia.org/wiki/Endo_rule#Endo_addition_rule endo rule]<ref name="Alder">K. Alder, G. Stein, ''Angew. Chem'', 1937, '''50''', 514: {{DOI|10.1002/ange.19370502804}}</ref> through the "maximum accumulation of unsaturated centers", which leads to a lower energy [http://en.wikipedia.org/wiki/Transition_state transition state] and hence a more rapid reaction.

This selectivity is also explained by [http://en.wikipedia.org/wiki/Woodward–Hoffmann_rules Woodward-Hoffmann rules] and secondary orbital interactions<ref name="Hoffmann">R. Hoffmann, R.B. Woodward, ''J. Am. Chem. Soc.'', 1965, '''87''', 4388: {{DOI|10.1021/ja00947a033}}</ref> for [http://en.wikipedia.org/wiki/Pericyclic_reaction pericyclic reactions], as shown in the diagram below.

[[Image:Secorb.JPG|500px]]

In both TS, the primary orbital overlap is the bonding interaction between the termini of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=C</sub> of the LUMO in maleic acid. This results in the formation of two σ bonds between the terminis of the two π systems.

However, in the ''endo'' TS, there exists another favourable bonding interaction. This is between the center carbons of the conjugated π system in the HOMO of cyclo-1,3-hexadiene and the π*<sub>C=O</sub> of the LUMO in maleic acid, thus stabilising the ''endo'' TS.

[[Image:DA2_endo_HOMO.jpg|170px|right]]This secondary orbital interactions<ref name="Hoffmann"/> can also be observed in the MO picture of the HOMO on the right. In the cyclo-1,3-hexadiene subunit, the two C=C bonds are out-of-phase with each other as observed by the nodal plane between the two center carbons of the conjugated system. In the maleic acid subunit, there is a nodal plane between the carbon of the C=C and the carbon of the C=O groups. Another nodal plane exists between the C and O of the C=O bond on the maleic acid subunit, indicating a π*<sub>C=O</sub> orbital. There is large overlap between the orbitals of the bond forming carbons, indicating a favourable bonding interaction.

There exists secondary orbital bonding interaction between the center carbon of the π conjugated system in the cyclo-1,3-hexadiene and the C atom of the π*<sub>C=O</sub> orbital since their orbitals have the same phase. On closer inspection, it is observed that the shared electron density between these orbitals passes from the top face of the π conjugated system through the back of the molecule, through the bottom face of the maleic acid subunit to the orbital on the C atom of the π*<sub>C=O</sub> orbital. This overlap stabilises the ''endo'' TS and thus lowers its activation energy.

In contrast, there is no such overlap observed in the ''exo'' TS, and thus the ''endo'' TS has a lower energy than the ''exo'' TS, so the kinetic product is expected to be the ''endo'' product.

As the [http://en.wikipedia.org/wiki/Diels_Alder Diels-Alder]<ref name="Sauer"/> reaction is irreversible, the kinetic ''endo'' product dominates. Hence, it can be concluded that the cycloaddition of cyclohexa-1,3-diene and maleic anhydride is kinetically controlled.

= Further Discussion =

What effects have been neglected in these calculations of Diels Alder transition states?

Look at published examples and investigate further if you have time. (e.g. DOI:10.1021/jo0348827 )

= References =
<references/>