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Rep:Mod:TRANJS8812

2015-03-27T11:01:05Z

Js8812: /* Optimising the Boat Transition State */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum, using Gaussian.<ref>Gaussian 09, Revision D.01, 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, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009.</ref>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Failed 'boat' optimisation</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Successful boat optimimsation</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>Diels, O. and Alder, K. (1928), Synthesen in der hydroaromatischen Reihe. Justus Liebigs Ann. Chem., 460: 98–122. doi: 10.1002/jlac.19284600106
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>Influence of steric interactions on endo stereoselectivity, J. Am. Chem. Soc., 1971, 93, ''18'' 4606</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

In fact, the endo product predominates, even though it is higher in energy due to less favourable interactions, as it is formed faster.

===Conclusion===
In conclusion, the calculations point towards the exo transition state as the more stable transition state - leading to the exo product. The program however doesn't consider the fact that the kinetic product can be formed faster than the thermodynamic product.

==References==
<references />

Rep:Mod:TRANJS8812

2015-03-27T10:54:16Z

Js8812: /* Molecular Orbitals of the Diels-Alder Products */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum, using Gaussian.<ref>Gaussian 09, Revision D.01, 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, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009.</ref>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>Diels, O. and Alder, K. (1928), Synthesen in der hydroaromatischen Reihe. Justus Liebigs Ann. Chem., 460: 98–122. doi: 10.1002/jlac.19284600106
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>Influence of steric interactions on endo stereoselectivity, J. Am. Chem. Soc., 1971, 93, ''18'' 4606</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

In fact, the endo product predominates, even though it is higher in energy due to less favourable interactions, as it is formed faster.

===Conclusion===
In conclusion, the calculations point towards the exo transition state as the more stable transition state - leading to the exo product. The program however doesn't consider the fact that the kinetic product can be formed faster than the thermodynamic product.

==References==
<references />

Rep:Mod:TRANJS8812

2015-03-27T10:49:01Z

Js8812: /* Conclusion */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum, using Gaussian.<ref>Gaussian 09, Revision D.01, 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, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009.</ref>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>Diels, O. and Alder, K. (1928), Synthesen in der hydroaromatischen Reihe. Justus Liebigs Ann. Chem., 460: 98–122. doi: 10.1002/jlac.19284600106
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>Influence of steric interactions on endo stereoselectivity, J. Am. Chem. Soc., 1971, 93, ''18'' 4606</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
In conclusion, the calculations point towards the exo transition state as the more stable transition state - leading to the exo product. The program however doesn't consider the fact that the kinetic product can be formed faster than the thermodynamic product.

==References==
<references />

Rep:Mod:TRANJS8812

2015-03-27T10:44:50Z

Js8812: /* The Diels-Alder Cycloaddition */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum, using Gaussian.<ref>Gaussian 09, Revision D.01, 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, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009.</ref>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>Diels, O. and Alder, K. (1928), Synthesen in der hydroaromatischen Reihe. Justus Liebigs Ann. Chem., 460: 98–122. doi: 10.1002/jlac.19284600106
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>Influence of steric interactions on endo stereoselectivity, J. Am. Chem. Soc., 1971, 93, ''18'' 4606</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

==References==
<references />

Rep:Mod:TRANJS8812

2015-03-27T10:42:23Z

Js8812: /* Optimisation of an Antiperiplanar 1,5-hexadiene Conformation */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum, using Gaussian.<ref>Gaussian 09, Revision D.01, 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, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian, Inc., Wallingford CT, 2009.</ref>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>{{Cite doi|10.1002/jlac.19284600106}}
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>{{cite journal|journal=J. Am. Chem. Soc. |year=1971|volume=93|pages= 4606|doi=10.1021/ja00747a052|title=Influence of steric interactions on endo stereoselectivity|last1=Houk|first1=K. N.|last2=Luskus|first2=L. J.|issue=18}}</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

==References==
<references />

Rep:Mod:TRANJS8812

2015-03-27T10:40:51Z

Js8812: /* References */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>{{Cite doi|10.1002/jlac.19284600106}}
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>{{cite journal|journal=J. Am. Chem. Soc. |year=1971|volume=93|pages= 4606|doi=10.1021/ja00747a052|title=Influence of steric interactions on endo stereoselectivity|last1=Houk|first1=K. N.|last2=Luskus|first2=L. J.|issue=18}}</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

==References==
<references />

Rep:Mod:TRANJS8812

2015-03-27T10:39:53Z

Js8812: /* Conclusion */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>{{Cite doi|10.1002/jlac.19284600106}}
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>{{cite journal|journal=J. Am. Chem. Soc. |year=1971|volume=93|pages= 4606|doi=10.1021/ja00747a052|title=Influence of steric interactions on endo stereoselectivity|last1=Houk|first1=K. N.|last2=Luskus|first2=L. J.|issue=18}}</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

==References==
{{Reflist}}

Rep:Mod:TRANJS8812

2015-03-27T10:39:19Z

Js8812: /* References */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>{{Cite doi|10.1002/jlac.19284600106}}
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>{{cite journal|journal=J. Am. Chem. Soc. |year=1971|volume=93|pages= 4606|doi=10.1021/ja00747a052|title=Influence of steric interactions on endo stereoselectivity|last1=Houk|first1=K. N.|last2=Luskus|first2=L. J.|issue=18}}</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

</ references>

==References==
{{Reflist}}

Rep:Mod:TRANJS8812

2015-03-27T10:38:01Z

Js8812: /* Introduction */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.<ref>{{Cite doi|10.1002/jlac.19284600106}}
</ref>
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. If the dienophile is unsymmetrical structure, there are two types of products that may form, exo-products and endo-products.<ref>{{Cite doi|10.1021/ed074p582}}</ref> Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster.<ref>{{cite journal|journal=J. Am. Chem. Soc. |year=1971|volume=93|pages= 4606|doi=10.1021/ja00747a052|title=Influence of steric interactions on endo stereoselectivity|last1=Houk|first1=K. N.|last2=Luskus|first2=L. J.|issue=18}}</ref> Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

==References==
{{Reflist}}

Rep:Mod:TRANJS8812

2015-03-27T10:32:06Z

Js8812: /* Optimising the Chair Transition Structure */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments. (Note: This is not the accepted intermediate for this reaction<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185–1204.</ref> but its geometry is highly useful in optimisations).

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

==References==
{{Reflist}}

Rep:Mod:TRANJS8812

2015-03-27T10:30:28Z

Js8812: /* Pericyclic Reactions */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes<ref>[[Arthur C. Cope]]; ''et al.''; ''[[J. Am. Chem. Soc.]]'' '''1940''', ''62'', 441.</ref>. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

==References==
{{Reflist}}

Rep:Mod:TRANJS8812

2015-03-27T10:29:14Z

Js8812: /* Molecular Orbitals of the Diels-Alder Products */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

===Conclusion===
Meow.

==References==
{{Reflist}}

Rep:Mod:TRANJS8812

2015-03-27T10:28:50Z

Js8812: /* The Cope Rearrangement */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

===Conclusion===
Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

==References==
{{Reflist}}

Rep:Mod:TRANJS8812

2015-03-27T10:28:21Z

Js8812:

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

==References==
{{Reflist}}

Rep:Mod:TRANJS8812

2015-03-27T10:25:39Z

Js8812: /* Chair Conformation IRC */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.692 a.u. after 44 steps, so it was not necessary to recompute another IRC.

[[File:SdChairIRC.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Eggbert.gif|thumb|center|225px|Animation of 'transition state']]

The IRC calculation did not converge to a transition state; for the final step, there is no imaginary vibrational frequency (the lowest value of the vibrational frequency is 61.84 cm<sup>-1</sup>.)

=====Boat Conformation IRC=====
The previously optimised boat was used to perform an IRC calculation, 65 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

The energy was found to minimise to -231.686 a.u. after 45 steps.

[[File:BoatIRCbyJaces.png|thumb|center|450px|IRC graph showing the reduction in total energy through calculation steps]]
[[File:Bonnieclyde.png|thumb|center|225px|Animation of transition state]]

Here a transition state had been reached; the imaginary vibrational frequency has a wavenumber of -150.83 cm<sup>-1</sup>

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

Rep:Mod:TRANJS8812

2015-03-27T10:17:45Z

Js8812: /* Molecular Orbitals of the Diels-Alder Products */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

The energies of the molecular orbitals are lower for the exo product than the endo product. All-in-all, through these optimisations and calculations, the energy of the exo product is calculated to be lower in energy than the endo product.

So why does the endo product predominate?

Rep:Mod:TRANJS8812

2015-03-27T10:16:19Z

Js8812: /* Molecular Orbitals of the Diels-Alder Products */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| -0.02649 a.u.
|}</div></center>

Rep:Mod:TRANJS8812

2015-03-27T10:15:49Z

Js8812: /* HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====Molecular Orbitals of the Diels-Alder Products====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| Symmetric
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| Anti-Symmetric
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| Symmetric
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| Anti-Symmetric
| -0.02649 a.u.
|}</div></center>

Rep:Mod:TRANJS8812

2015-03-27T10:14:41Z

Js8812: /* HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| Symmetric
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| Anti-Symmetric
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMObeta.png|300px]]
| Symmetric
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMObeta.png|300px]]
| Anti-Symmetric
| -0.02649 a.u.
|}</div></center>

Rep:Mod:TRANJS8812

2015-03-27T10:13:09Z

Js8812: /* HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Endo Product HOMO
| [[File:Endo631HOMO.png|300px]]
| Symmetric
| -0.23810 a.u.
|-
| Endo Product LUMO
| [[File:Endo631LUMO.png|300px]]
| Anti-Symmetric
| -0.02076 a.u.
|-
| Exo Product HOMO
| [[File:Exo631HOMOBeta.png|300px]]
| Symmetric
| -0.27495 a.u.
|-
| Exo Product LUMO
| [[File:Exo631LUMOBeta.png|300px]]
| Anti-Symmetric
| -0.02649 a.u.
|}</div></center>

Rep:Mod:TRANJS8812

2015-03-27T10:06:22Z

Js8812: /* Optimisation of Cyclohexa-1,3-diene */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| 0.02771129 a.u.
| -233.41891660 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| 0.00000664 a.u.
|-
| '''Dipole Moment'''
| 0.4313 Debye
| 0.3777 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

Rep:Mod:TRANJS8812

2015-03-27T10:03:34Z

Js8812: /* Optimisation of Maleic Anhydride */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.12182418 a.u.
| -379.28953540 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| 0.00003415 a.u.
|-
| '''Dipole Moment'''
| 4.5779 Debye
| 4.0720 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

Rep:Mod:TRANJS8812

2015-03-27T09:59:48Z

Js8812: /* Towards Cyclohexene - The Diels-Alder Transition State */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

The LUMO for this transition state is essentially very similar to the LUMO of s-cis-butadiene (though it is slightly lower in energy, due to slightly favourable interactions between the MOs of ethylene).

The HOMO of this transition state is the combination of the HOMO of the cis-butadiene and the LUMO of the ethylene.

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

Rep:Mod:TRANJS8812

2015-03-27T09:52:51Z

Js8812: /* Towards Cyclohexene - The Diels-Alder Transition State */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Transition State HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Transition State LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

Rep:Mod:TRANJS8812

2015-03-27T09:51:44Z

Js8812: /* Towards Cyclohexene - The Diels-Alder Transition State */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product. The ethylene is now optimised to be closer to the butadiene, with a distance of 1.48 Å (slightly longer than the σ C-C bonds of butadiene itself.)

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Anti-Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:LUMALEE.png|300px]]
| Symmetric
| 0.01693 a.u.
|}</div></center>

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

File:LUMALEE.png

2015-03-27T09:50:36Z

Js8812: ŌŌŌŌŌ

ŌŌŌŌŌ

Rep:Mod:TRANJS8812

2015-03-27T09:44:38Z

Js8812: /* Ethylene Optimisation and MO Visualisation */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Ethylene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Ethylene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

Rep:Mod:TRANJS8812

2015-03-27T09:43:59Z

Js8812: /* Ethylene Optimisation and MO Visualisation */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| -0.38775 a.u.
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| 0.05283 a.u.
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

Rep:Mod:TRANJS8812

2015-03-27T09:42:35Z

Js8812: /* Chair Conformation IRC */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]
[[File:Bonnieclyde.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| ????
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| ????
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

File:Bonnieclyde.png

2015-03-27T09:42:11Z

Js8812: homeőőőőőőő

homeőőőőőőő

Rep:Mod:TRANJS8812

2015-03-27T09:38:03Z

Js8812: /* Chair Conformation IRC */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

[[File:BoatIRCbyJaces.png]]

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| ????
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| ????
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

File:BoatIRCbyJaces.png

2015-03-27T09:37:05Z

Js8812: —————Ů

—————Ů

Rep:Mod:TRANJS8812

2015-03-27T03:45:52Z

Js8812: /* Maleic Acid and Cyclohexadiene: A Study in Regioselectivity */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| ????
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| ????
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

====Optimisation of Maleic Anhydride====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

====Optimisation of Cyclohexa-1,3-diene====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

====The Endo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>EndoEndo.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised endo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Endo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387277 a.u.
| -613.70918462 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001592 a.u.
| 0.03275780 a.u.
|-
| '''Dipole Moment'''
| 5.5057 Debye
| 7.8200 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====The Exo Product Optimisation====
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>ExoAM!.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised exo product</title><color>darkslategray</color><size>250</size><uploadedFileContents>Exo631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Calculation Method'''
| RAM1
| RB3LYP
|-
| '''Basis Set'''
| ZDO
| 6-31G(D)
|-
| '''Total Energy'''
| -0.21387281 a.u.
| -613.99106865 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001101 a.u.
| 0.00006005 a.u.
|-
| '''Dipole Moment'''
| 5.5059 Debye
| 4.9382 Debye
|-
| '''Point Group'''
| C<sub>s</sub>
| C<sub>s</sub>
|}</div></center>

====HOMO AND LUMO OF THE MALEIC CRAZINESS KITTY!!! ====
[[File:Endo631HOMO.png|thumb|300px|ENDO HOMO]][[File:Endo631LUMO.png|thumb|300px|ENDO LUMO]][[File:Exo631HOMObeta.png|thumb|300px|EXO HOMO]][[File:Exo631LUMObeta.png|thumb|300px|EXO LUMO]]

all the boring discussion rubbish will be vomited out tomorrow :)

File:Exo631LUMObeta.png

2015-03-27T03:44:59Z

Js8812: i feel a clld flush

i feel a clld flush

File:Exo631HOMObeta.png

2015-03-27T03:44:18Z

Js8812: dsdsdsd

dsdsdsd

File:Endo631LUMO.png

2015-03-27T03:43:39Z

Js8812:

File:Endo631HOMO.png

2015-03-27T03:42:48Z

Js8812:

File:Exo631.mol

2015-03-27T03:40:21Z

Js8812: fd

File:ExoAM!.mol

2015-03-27T03:39:39Z

Js8812: δ

File:Endo631.mol

2015-03-27T03:35:57Z

Js8812: f

File:EndoEndo.mol

2015-03-27T03:35:11Z

Js8812:

Rep:Mod:TRANJS8812

2015-03-27T03:33:03Z

Js8812: /* Maleic Acid and Cyclohexadiene: A Study in Regioselectivity */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| ????
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| ????
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

This transition structure is first optimised using the Semi-Empirical/AM1 level of theory, then further optimised using the more sophisticated B3LYP/6-31G* level of theory for comparison. There is a noticeable difference in the time taken to perform the optimisation calculations between the almost instantaneous AM1 method and the longer B3LYP method.

MALEIC ANHYDRIDE
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised maleic anhydride</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleicanhydrideam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised maleic acid</title><color>darkslategray</color><size>250</size><uploadedFileContents>Maleic631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| -0.12182418 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00003683 a.u.
| ????
|-
| '''Dipole Moment'''
| 4.5779 Debye
| ????
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

CYCLOHEXADIENE
<center><div>
{| class="wikitable"
|
|'''Semi-Empirical/AM1'''
|'''B3LYP/6-31G*'''
|-
|'''Visual'''
|<jmol><jmolApplet><title>AM1 optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadieneam1.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>6-31G* optimised cyclohexa-1,3-diene</title><color>darkslategray</color><size>250</size><uploadedFileContents>Cyclohexadiene631.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
| ????
|-
| '''Calculation Method'''
| RAM1
| ????
|-
| '''Basis Set'''
| ZDO
| ????
|-
| '''Total Energy'''
| 0.02771129 a.u.
| ????
|-
| '''RMS Gradient Norm'''
| 0.00000562 a.u.
| ????
|-
| '''Dipole Moment'''
| 0.4313 Debye
| ????
|-
| '''Point Group'''
| C<sub>2</sub>
| C<sub>2</sub>
|}</div></center>

File:Cyclohexadieneam1.mol

2015-03-27T03:31:09Z

Js8812:

File:Cyclohexadiene631.mol

2015-03-27T03:29:36Z

Js8812:

File:Maleicanhydrideam1.mol

2015-03-27T03:20:50Z

Js8812:

File:Maleic631.mol

2015-03-27T03:19:00Z

Js8812: 631

631

Rep:Mod:TRANJS8812

2015-03-27T03:15:40Z

Js8812:

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| ????
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| ????
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===
YOU KNOW YOU SAID ITS TRUE CAN YOU FEEL THE LOVEI CAN FEEL IT TO>

Rep:Mod:TRANJS8812

2015-03-27T03:14:49Z

Js8812: /* The Diels-Alder Cycloaddition */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| ????
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| ????
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene: A Study in Regioselectivity===

Rep:Mod:TRANJS8812

2015-03-27T03:14:18Z

Js8812: /* The Diels-Alder Cycloaddition */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| ????
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| ????
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

oh boy

===Maleic Acid and Cyclohexadiene - A Study in Regioselectivity===

Rep:Mod:TRANJS8812

2015-03-27T03:12:00Z

Js8812: /* The Diels-Alder Cycloaddition */

== The Cope Rearrangement ==

=== Introduction ===
==== Pericyclic Reactions ====
Pericyclic reactions are highly useful and versatile rearrangment methods in organic synthesis. The appeal of these reactions is the lack of need of extra reagents and specific solvents; only heat or electromagnetic light is required to drive the reaction. These reactions are also useful for forming rings from unsaturated compounds.

One useful example of a pericyclic reaction is the Cope Rearrangment of 1,5-dienes. This is a [3,3]-sigmatropic rearrangement which occurs on heating the diene.

<center><div>
{|class="wikitable" border="1" cellpadding="4" cellspacing="4"
|[[File:JDSCope_rearrangement.jpg]]
|-
|''Scheme 1'': Mechanism of Cope Rearrangement
|}
</div></center>

In this example, the Cope Rearrangement can progress via a chair or boat transition state, based on the conformations of cyclohexanes.

Computational methods will be used to optimise the reactants and the transition states, using the Hartree-Fock method of calculation or the B3LYP method of calculation.

===1,5-hexadiene Conformations===

1,5-hexadiene can exist in several different conformations, dependent on the four central atoms being gauche or antiperiplanar.

====Optimisation of an Antiperiplanar 1,5-hexadiene Conformation====

A molecule of 1,5-hexadiene was drawn, with the four central carbon atoms all in the anti-periplanar conformation. Using the Hartree-Fock model and the 3-21G basis set, the drawn molecule was optimised to a minimum.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69253528 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
|}</div></center>

The calculation was successful and the molecule was found to possess a center of inversion.

==== Optimisation of a Gauche 1,5-hexadiene Conformation ====

This time, a molecule of 1,5-hexadiene was drawn, but with the four central carbon atoms in a ''gauche'' conformation. Again, using the Hartree-Fock model and the 3-21G basis set, this molecule was optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>A gauche conformation of 1,5-hexadiene (gauche4)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGaucheHexa.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69153032 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001866 a.u.
|-
| '''Dipole Moment'''
| 0.1281 Debye
|-
| '''Point Group'''
| C<sub>2</sub>
|}</div></center>

As expected, the minimum energy of the gauche conformation is higher than the minimum energy for the anti conformation drawn. It is expected that there is a more repulsive force between the methylene groups in the gauche conformation than in the anti conformation, and steric hindrance from flagpole interactions between hydrogen atoms on adjacent carbon atoms.

==== The Most Thermodynamically Stable Conformer ====

The most stable (lowest energy) conformation of 1,5-hexadiene is predicted to be an anti conformation, as ''generally'', anti conformers are more stable than gauche conformers due to less steric hindrance and less unfavourable interactions between neeigbouring protons.

The ''anti1'' conformation was found to be the most stable of all the anti conformers.

In fact, there is a gauche conformation which is more stable than any of the anti conformers. There is less steric hindrance and less pronounced flagpole interatcions, decreasing any steric hindrance, making this conformer the most stable overall.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Minimum energy conformation of 1,5-hexadiene (gauche3)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSGauche3.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.69266120 a.u
|-
| '''RMS Gradient Norm'''
| 0.00001535 a.u.
|-
| '''Dipole Moment'''
| 0.3409 Debye
|-
| '''Point Group'''
| C<sub>1</sub>
|}</div></center>

The value of -231.69266120 for the total internal energy of the molecule is the lowest of all conformers of 1,5-hexadiene. This is the ''gauche3'' conformation.

===Comparing the Hartree-Fock/3-21G and B3LYP/6-31G* Levels of Theory===

There are many types of theories, calculation methods and orbital basis sets that can be used to fine-tune and ameliorate the minimum optimisations. Investigated below is the comparison between the Hartree-Fock/3-21G and B3LYP/6-31G* levels of theory and basis sets.

The first confomer optimised, the ''anti2'' confomer, was reoptimised to a minimum using the B3LYP method and the 6-31G* basis set. Below is a comparison of values between the two results.

<center><div>
{| class="wikitable"
|
|<jmol><jmolApplet><title>An anti conformation of 1,5-hexadiene (anti2)</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSExerciseA.mol</uploadedFileContents></jmolApplet></jmol>
|<jmol><jmolApplet><title>An anti conformation calculated using B3LYP/6-31G* methods.</title><color>darkslategray</color><size>250</size><uploadedFileContents>2ANTIHEXA.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Method/Basis Set
| HF/3-21G
| B3LYP/6-31G*
|-
| '''Calculation Type'''
| FOPT
| FOPT
|-
| '''Total Energy'''
| -231.69253528 a.u
| -234.61171063 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001891 a.u.
| 0.00001249 a.u.
|-
| '''Dipole Moment'''
| 0.0000 Debye
| 0.0000 Debye
|-
| '''Point Group'''
| C<sub>i</sub>
| C<sub>i</sub>
|}</div></center>

Observed is a further reduction in energy using the B3LYP/6-31G* level of theory, meaning that the 6-31G* orbital basis set is a slightly more effective in optimisations. There is a reduction in energy of 2.91917535 a.u. and a reduction in the RMS gradient norm. There is, however, no visible difference in structure and geometry between both results.

===A More Detailed Study of the Energy of 1,5-hexadiene===
The energy calculated via Gaussian is the energy of the molecule according to the potential energy surface. Using a molecule optimised to a minimum using the B3LYP/6-31G method, a frequency calculation can be performed on this molecule. Four different types of energies are ultimately calculated:

[[File:2HexaAnti_Infrared_Vibration_13.png|thumb|250px|left|Simulation of IR vibration spectrum, highlighting maximum peak - vibrational mode #13]]

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' (E<sub>1</sub> = E<sub>elec</sub> + E<sub>zp</sub>)</li>
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' (E<sub>2</sub> = E<sub>elec</sub> + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)</li>
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' (H = E + RT)</li>
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' (G = H - TS)</li>
</ol>

Using the previuosly B3LYP/6-31G* optimised ''anti2'' confomer of 1,5-hexadiene, a frequency calculation was performed, yielding a list of vibration frequencies. The calculation was confirmed to be successful, as there were no negative (imaginary) vibrational frequencies present.

An IR spectrum of all vibrational modes was simulated, showing major peaks at 940.00 cm<sup>-1</sup> - the highest peak (vibrational mode #13), 3030.78 cm<sup>-1</sup> - (vibrational mode #34) and 3136.02 cm<sup>-1</sup> - (vibrational mode #38).

[[File:2HexaAnti_Vibraions.png|thumb|249px|right|List of vibrations from the frequency calculation of 1,5-hexadiene ''anti2'']]

As a result of this calculation the values for all aforementioned energies were determined, at 298.15 K.

<ol>
<li>'''E<sub>1</sub>: Sum of Electronic and Zero-Point Energies''' = -234.469219 a.u.
<li>'''E<sub>2</sub>: Sum of Electronic and Thermal Energies''' = -234.461869 a.u.
<li>'''E<sub>3</sub>: Sum of Electronic and Thermal Enthalpies''' = -234.460925 a.u.
<li>'''E<sub>4</sub>: Sum of Electronic and Thermal Free Energies''' = -234.500809 a.u.
</ol>

The lowest value of energy from this list is the sum of the electronic and thermal free energy.

===Optimising the Chair and Boat Transition Structures===

The chair and boat transition structures of these reactions are based on the respective conformations of cyclohexane. Here these structures are predicted, and further optimised.

====Optimising the Chair Transition Structure====
The 'chair' conformation of the Cope transition structure can be shown as a biradical transition state, with two specifically positioned allyl radical fragments.

[[File:JacquesSmithChair.jpg|thumb|left|Biradical 'chair' transition state of Cope rearrangement]]

To begin with optimising the 'chair' conformation of the cope transition structure, the allyl fragment itself must first be optimised. <br><br><br><br><br><br><br><br><br><br><p>

=====Allyl Fragment Optimisation=====

An allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimised to a minimum using the HF/3-21G level of theory.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSAllylFrag.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| UHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -115.82303830 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00018761 a.u.
|-
| '''Dipole Moment'''
| 0.0299 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Notable is the fact that the total energy of the optimised allyl fragment is close to half of the value of the total energy for the previously optimised 1,5-hexadiene confomers.

=====A Pair of Allyls - The Next Step Towards Optimising the 'Chair'=====

[[File:PartyInTheUsa.png|thumb|left|160px|Chair transition state guess]]

Two allyl separate allyl fragments were drawn, positioning them so they are similar in appearance to the chair conformation of cyclohexane. The distance between terminal carbons of separate allyl fragments were guessed to be 2.2 Å.

This prototype structure was then optimised straight away. In this optimisation, a frequency calculation was included, and the structure was optimised, this time to a transition state, using the Berny algorithm. The Hessian force constants were calculated just once throughout.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932237 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003062 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Exercisebanim.gif|220px|thumb|right|Animation of the imaginary vibrational frequency of the optimised transition state]]

Observed is one imaginary vibrational frequency, at -818.04 cm<sup>-1</sup>. An imaginary vibrational frequency is associated with a transition state, here in the case of the intermediate formation of new C-C σ-bonds. The

This optimised structure has a slightly higher value of total energy than twice the total energy of a single allyl fragment. This is expected, due to the extra interactions between the allyl fragments.

Notable in this optimised transition state is that the distance between terminal carbon atoms of separate allyl fragments were reduced to 2.02 Å, which is overall undesired. To counter this result during the optimisation, atoms and bonds can be frozen in place during the calculations, and this was taken into account in the next optimisation.

The distances between terminal carbons of separate allyl fragments were both frozen at 2.2 Å, allowing just the central carbon atoms of each allyl fragment to be free to move during the optimisation. This allows a more precise and fine-tuned optimisation overall.

Once the atoms were frozen, this time, the transition state was optimised to a minimum, using the HF/3-21G level of theory, as the transition state structure was already optimised.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised to transition state: allyl fragment pair</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSExeanim.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61346000 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00558446 a.u. a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

Observed is a slight increase in total energy, due to the increase in bond length from 2.02 Å from the previously optimised transition state to 2.2 Å.

[[File:JDSAnimExerC.gif|thumb|200px|right|Vibrational animation of the transition state optimised through freezing atom positions]]

The next step for the complete optimisation of the 'chair' conformation transition state is to optimise the distances between the allyl fragments. This was done by incorporating the derivative option within the redundant co-ordinates, for both inter-fragment distances. The structure was then optimised to a transition state via the Berny algorithm.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Fully optimised chair transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>EXERD.mol
</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FTS
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61518500 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>

[[File:Animexerd.gif|thumb|200px|right|Animation of final optimised 'chair' transition state]]
As a result of this optimisation, there is a further reduction in total energy. The distance between fragments of the terminal carbon atoms is now 2.197 Å, and the distance between the central carbon atoms has increased from 2.88 Å to 2.94 Å. The C<sub>1</sub>-C<sub>2</sub>-C<sub>3</sub> angle has as a result reduced from 122.9° to 121.9°.

====Optimising the Boat Transition State====
The 'boat' transition state is the other possible transition state that the Cope rearrangement can progress through. It is expected to be higher in energy than the chair transition state, mirroring the energy profiles of the different conformers of cyclohexane.

The transition state is not based on allyl fragments; instead the original 1,5-hexadiene molecule and the desired product can be drawn, which is used to generate a transition state.

To begin, the ''anti2'' conformer of 1,5-hexadiene was drawn. This particular rearrangement will yield another molecule of 1,5-hexadiene, so this molecule was duplicated and used as the product for the calculation.

To discern between the reacting molecule and the final product, it is essential to label each carbon atom, and take into account their position in the product molecule. The atoms for this calculation were therefore labelled as such:

{| class="wikitable"
|-
|[[File:JDSEE1.png|320px|centre]]
|[[File:JDSEE2.png|320px|centre]]
|}

There is now enough information for an optimisation to a transition state, using the QST2/QST3 method. The QST2 method was chosen for this calculation; (the QST3 allows for the possibility of including a 'guess' transition state but this is not required here.)

The calculation completed without any errors, but the transition state did not have any geometry with resemblance to a boat.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>Badboat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.61932232 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003274 a.u.
|-
| '''Dipole Moment'''
| 0.0003 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
|}</div></center>
[[File:Transitionstatefialureanimated.gif|right|200px|thumb|Animation of the unsuccessful optimisation]]

The calculation produces a transition state resembling a chair, but there are interactions between opposite ends of the allyl fragments. This transition state has an imaginary vibrational frequency of -817.91cm<sup>-1</sup>, similar to the 'guess' transition structure for the 'chair' conformation.

The reason this calculation fails is because it does not consider the possibility of the central carbon atoms rotating. As shown in the mechanism for the Cope rearrangement, the reaction occurs when the terminal carbon atoms are closer to each-other, in effect, almost forming a complete ring. A straighter chain will not allow the formation of a new σ C-C bond as the two carbon atoms will be too far away to interact. This calculation does not take into account the fact that the central carbon atoms can rotate around to accommodate the possibility of the sigmatropic rearrangement. It is therefore necessary to manually re-orient the reactant and product molecule so that they are more 'C'-shaped.

The reactant and product molecules were then re-orientated by reducing the dihedral angle of the four central carbon atoms from 180° to 0°. The C<sup>2</sup>-C<sup>3</sup>-C<sup>4</sup> and C<sup>3</sup>-C<sup>4</sup>-C<sup>5</sup> angles were reduced from 109° to 100°, as shown in the corresponding table:

{| class="wikitable"
|-
|[[File:Ee3.png|320px|centre]]
|[[File:Ee4.png|320px|centre]]
|}

[[File:JDSBoatanimated.gif|thumb|200px|right|Animation of the vibration of the boat transition state]]

These conformations of 1,5-hexadiene look much more likely to re-arrange and it is therefore more likely that the calculation will successfully yield the boat transition state.

The transition structure optimisation was performed, again using the QST2 method.

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimised allyl fragment</title><color>darkslategray</color><size>250</size><uploadedFileContents>JDSBoat.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FREQ
|-
| '''Calculation Method'''
| RHF
|-
| '''Basis Set'''
| 3-21G
|-
| '''Total Energy'''
| -231.60280200 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The boat structure was successfully optimised, with the expected C<sub>2v</sub> point group.
There is indeed only one vibration frequency of -839.94 cm<sub>-1</sub>, more negative than the previous failed transition state calculated.

====Tracking the Optimisation with Intrinsic Reaction Coordinates====

It is useful to track the steps of the optimisation. A method of performing this in Gaussian is to run an Intrinsic Reaction Coordinate (IRC) calculation. The calculation then plots the minimum energy path throughout an optimisation, and the resulting graph will level off to a minimum energy (if the number of steps allows it to). The IRC calculation will also aid in predicting the 1,5-hexadiene conformer product.

=====Chair Conformation IRC=====
The previously optimised chair was used to perform an IRC calculation, 50 steps being specified. The calculation was only computed in the forward direction. The force constants were also computed for each step.

[[File:SdChairIRC.png]]
[[File:Eggbert.gif]]

CHAIR: 61.84 cm-1

BOAT: -150.83 cm-1

Not successful.

====Boats, Chairs, Basis Sets and Energies====
The 'chair' and 'boat' transition states were previously optimised using the Hartree-Fock/3-21G level of theory. It was previously concluded, through the optimisations of 1,5-hexadienes, that the B3LYP/6-31G* level of theory is more effective at lowering the total energy of a molecule. The re-optimised transition states can be further used to obtained more accurate values for their intrinsic energy properties, more accurately than using the HF/3-21G level of theory.

It is in fact useful to use a lower level of theory to obtain a good prototype optimisation structure to work from, higher levels of theory can be conducted more quickly and accurately afterwards.

The 'chair' and 'boat' transition state were therefore re-optimised using the B3LYP/6-31G* level of theory.

=====Comparison of Chair Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.61518500 a.u.
| -232.98235323 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00326133 a.u.
| 0.00000760 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
| 0.3960 Debye
|-
| '''Point Group'''
| C<sub>2h</sub>
| C<sub>2h</sub>
|}</div></center>

There is an expected reduction of total energy through this re-optimisation. There is also no visible change in geometry.

There is, however an increase in dipole moment, which is an unexpected result. It has not been conjectured on why this drastic change in dipole moment has taken place.

=====Comparison of Boat Optimisations=====

<center><div>
{| class="wikitable"
| '''Method/Basis Set
| '''HF/3-21G'''
| '''B3LYP/6-31G*'''
|-
| '''Calculation Type'''
| FREQ
| FREQ
|-
| '''Total Energy'''
| -231.60280200 a.u.
| -234.54045615 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00007080 a.u.
| 0.00488727 a.u.
|-
| '''Dipole Moment'''
| 0.1579 Debye
| 0.0562 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
| C<sub>2v</sub>
|}</div></center>

The energy difference for this reoptimisation is more drastic than in the reoptimisation for the 'chair' conformation. Through the HF/3-21G basis set, it was the 'chair' conformation that had the lowest value of total energy. After the reoptimsations with B3LYP/6-31G* basis set, the 'boat' conformation is now lower in energy than the 'chair' conformation. The level of theory is therefore very important in certain calculations.

There is also a reduction in total dipole moment from HF/3-21G to B3LYP/6-31G*.

=====Activation Energies=====
The activation energies of both transition structures can be calculated from the optimisations. A value for the zero-point energy is calculated which is then used to work out the activation energy. Below is the comparison between experimental and optimisation:

<center><div>
{| class="wikitable"
| '''Conformation
| '''Experimental E<sub>a</sub> (0K)'''
| '''Optimisation E<sub>a</sub> (0K'''
|-
| Chair
| 33.5 ± 0.5 kcal/mol
| 33.561 kcal/mol
|-
| Boat
| 44.7 ± 2.0 kcal/mol
| 42.592 kcal/mol
|}</div></center>

The values are relatively similar for the chair conformation, but drastically different for the boat conformation.

The energies were also calculated at 298.15 K:

<center><div>
{| class="wikitable"
| '''Conformation
| '''E<sub>a</sub> (298.15 K)'''
|-
| Chair
| 31.105 kcal/mol
|-
| Boat
| 41.822 kcal/mol
|}</div></center>

The energies reduce at a higher temperature - expected as atmospheric energy contributes more to the activation at a higher tempeature.

Through all this data, it is the chair transition state which this reaction proceeds via, as the chair transition state has the lowest activation energy, even though the boat transition state is more thermodynamically stable.

==The Diels-Alder Cycloaddition==

===Introduction===
The Diels-Alder cycloaddition is a synthesis classic - a highly versatile, simple pericyclic reaction which allows the formation of cyclohexene rings, which can be further reacted as part of long synthetic routes.
[[File:Dielsaldera.png|thumb|left|300px|Mechanism of Diels-Alder cycloaddition]]
The Diels-Alder cycloaddition is a [4+2]-cycloaddition, involving the formation of two new C-C σ-bonds. Two reactants are required, an s-cis-diene and a dienophile (which doesn't necessarily have to be a C=C double bond). To form the desired product, the reaction must be heated.
[[File:MaleicAnhydride.jpg|thumb|right|300px|An example of endo and exo products of Diels-Alder reactions]]

The simplest example is the reaction between s-cis butadiene and ethylene to form cyclohexene. As the reactants become more complicated in structure, it is apparent to see two types of product forming, exo-products and endo-products. Exo-products are thermodynamically more stable, as endo-products suffer from more severe steric effects, but usually the endo-product predominates in Diels-Alder cycloadditions, as it is the kinetic product and is formed faster. Refer to the diagram of the reaction between cyclohexa-1,3,-diene and maleic anhydride to view the structure of these products.

===Simple Diels-Alder Optimisations===
To understand the reaction through optimisations, the simplest example of a Diels-Alder reaction, the reaction between s-cis-butadiene and ethylene was modelled. To begin with, the semi-empirical AM1 method, lower in sophistication than the HF/3-21G model, was used for these optimisations.

====Ethylene and Butadiene====

=====S-Cis Butadiene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of s-cis-butadiene</title><color>darkslategray</color><size>250</size><uploadedFileContents>cisbutnomo.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.04879718 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00001429 a.u.
|-
| '''Dipole Moment'''
| 0.0414 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

Once optimised, the list of molecular orbitals can be calculated and visualised. The HOMO and LUMO are visualised as they are the most important in determining how the reaction with ethylene will take place.

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSCisbutHOMO.png|300px]]
| Anti-symmetric
| -0.34382 a.u.
|-
| Butadiene LUMO
| [[File:JSCisbutLUMO.png|300px]]
| Symmetric
| 0.01708 a.u.
|}</div></center>

=====Ethylene Optimisation and MO Visualisation=====
<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>Optimisation of ethylene</title><color>darkslategray</color><size>250</size><uploadedFileContents>JSEthene.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02619028 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00003649 a.u.
|-
| '''Dipole Moment'''
| 0 Debye
|-
| '''Point Group'''
| D<sub>2h</sub>
|}</div></center>

The HOMO and LUMO are then calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:JSEtheneHOMO.png|300px]]
| Symmetric
| ????
|-
| Butadiene LUMO
| [[File:JSEtheneLUMO.png|300px]]
| Anti-Symmetric
| ????
|}</div></center>

====Towards Cyclohexene - The Diels-Alder Transition State====
[[File:Noplaceidratherbe.png|left|thumb|320px|Diels Alder reaction including 'envelope' transition state]]
The Diels-Alder cycloaddition proceeds via an 'envelope' transition state, which will be optimised. This conformation is favourable as there is a large overlap between the π-systems of both reactants. The distance between the butadiene and ethylene is estimated at 2.0 Å, but the value is found via the optimised strucutre.
<br><br><br><br><br>

<center><div>
{| class="wikitable"
|<jmol><jmolApplet><title>6-31G* cyclohexene transition state</title><color>darkslategray</color><size>250</size><uploadedFileContents>DielsalderAM1.mol</uploadedFileContents></jmolApplet></jmol>
|-
| '''Calculation Type'''
| FOPT
|-
| '''Calculation Method'''
| RAM1
|-
| '''Basis Set'''
| ZDO
|-
| '''Total Energy'''
| 0.02795804 a.u.
|-
| '''RMS Gradient Norm'''
| 0.00004317 a.u.
|-
| '''Dipole Moment'''
| 0.4558 Debye
|-
| '''Point Group'''
| C<sub>2v</sub>
|}</div></center>

The envelope transition state drawn has unexpectedly been optimised into an almost-planar hexagon, and not the unexpected envelope; meaning that this optimisation may have gone beyond the transition state, and closer to the cyclohexene product.

The HOMO and LUMO are calculated and visualised:

<center><div>
{| class="wikitable"
|'''Molecular Orbital'''
|'''Visualisation'''
|'''Symmetry'''
|'''Energy of MO'''
|-
| Butadiene HOMO
| [[File:DielsalderHOMOSTOPSTEALINGMYNAME.png|300px]]
| Symmetric
| -0.32074 a.u.
|-
| Butadiene LUMO
| [[File:Thisove.gif|300px]]
| Anti-Symmetric
| 0.01693 a.u.
|}</div></center>

===Maleic Acid and Cyclohexadiene - A Study in Regioselectivity===